tag:blogger.com,1999:blog-35789873018141349712024-03-13T05:51:28.214+05:30The Dynamic FrequencyThe sole purpose behind the creation of this blog is to make the most difficult topics in Quantum physics,relativity and astronomy as simple as eating a piece of cake and to develop positive thinking about learning the subject. It will help our generation to understand the mechanism of our nature and eventually will help to discover more wonders. Not only that, understanding all those will help our generation to innovate hence, making the life of people easier than ever before. Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.comBlogger83125tag:blogger.com,1999:blog-3578987301814134971.post-73566055204548644102022-03-15T22:10:00.002+05:302022-03-15T22:25:54.800+05:30How to Land an Orbital Class Rocket...<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh8ndHddiXHQeQHTcK37oi-LNs1UQdWrJr7GNoLZTLRuPL9PBoM2850mHPYlDxytRkUA8hG9mziBSDkBHmHkDIMOI2YqpttKj9X8BnY-UP5JGeuVSRG-nlXiFQDFZcdb41QA3Mt94662OVVBhiiHOj_mXvSDpB122WXkp6yCNpqKFSTfJp5b3cRqRQT=s3000" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="2000" data-original-width="3000" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEh8ndHddiXHQeQHTcK37oi-LNs1UQdWrJr7GNoLZTLRuPL9PBoM2850mHPYlDxytRkUA8hG9mziBSDkBHmHkDIMOI2YqpttKj9X8BnY-UP5JGeuVSRG-nlXiFQDFZcdb41QA3Mt94662OVVBhiiHOj_mXvSDpB122WXkp6yCNpqKFSTfJp5b3cRqRQT=s320" width="320" /></a></div><p></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Hello
all, welcome to yet another blog where I’ll be explaining rocket science and
breaking the concepts into simple principles of physics. Today we shall get to
know how <b>SpaceX</b>’s <b>Falcon9</b> booster-stage lands with precision. But why only
Falcon9?? I mean <b>orbital rockets</b> have been in existence since the <b>1950s’</b> but
there had been no vertically landing rockets until <b>2015</b>. It may seem surprising
but the concept of vertical landings is pretty old. So, why we didn’t see any
of the vertical landing rockets? Hold on, this article will solve all your
doubts, so without much further ado let’s dive in headfirst.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Lift-off:<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhK1luCKYeLheYXpvAHcrVEf_iO5lOMR4F0cZtnbU08xtMXs9w45jE1W9IbCfebOi6--oCyVs-3DFx58E9coVE8Z4OBKcbA3lM-kX1xQWNCfpkrIJQKPcskYM0deEe3Z-pHZlduu_MXMJpiptUa6NlBSDPVbTjaSEIkPUqSCGsElU3qGgLsqfgeQMM7=s4358" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="4358" data-original-width="3648" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhK1luCKYeLheYXpvAHcrVEf_iO5lOMR4F0cZtnbU08xtMXs9w45jE1W9IbCfebOi6--oCyVs-3DFx58E9coVE8Z4OBKcbA3lM-kX1xQWNCfpkrIJQKPcskYM0deEe3Z-pHZlduu_MXMJpiptUa6NlBSDPVbTjaSEIkPUqSCGsElU3qGgLsqfgeQMM7=w268-h320" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="268" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Falcon 9 lifting off</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The Falcon9 at the time of lift-off stands at the height of <b>70m</b> and weighs about <b>550,000 kg</b>. The rocket lifts off as all <b>9 merlin
engines</b> are turned on. The booster burns for about <b>162s</b> before the engines are
turned off. We observe <b>MECO</b> (Main Engine Cut-Off) at about <b>72.3 km </b>of altitude
while the vehicle is traveling at about <b>7800 km/h</b>. The 2<sup>nd</sup> stage
Merlin engine is turned on after separation to achieve orbit. At the same time,
the booster stage turns around with the help of <b>cold gas thrusters/nitrogen
thrusters</b> and begins the <b>boost back burn</b>.</span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Boost back burn:<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi-rTrnF8YuLZw0EYiWuK0_vykMD7v173MBAtYw2oBCmEy6pF0vpSTb2BuklYe6PFG9ShDO0Ieu93vk2RS1xWk_CPidxypcdA44qOQP21bUaFi2albovmnx18CCIbDTIksPVcV_H0iVQa0jh76oATZHlsYolW_l0WGYrw71tsL9gn-k06Sfal2erHgA=s996" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="812" data-original-width="996" height="261" src="https://blogger.googleusercontent.com/img/a/AVvXsEi-rTrnF8YuLZw0EYiWuK0_vykMD7v173MBAtYw2oBCmEy6pF0vpSTb2BuklYe6PFG9ShDO0Ieu93vk2RS1xWk_CPidxypcdA44qOQP21bUaFi2albovmnx18CCIbDTIksPVcV_H0iVQa0jh76oATZHlsYolW_l0WGYrw71tsL9gn-k06Sfal2erHgA=w320-h261" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Booster preparing to land</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">There is a certain amount of <b>energy</b> required to get
the<b> 2<sup>nd</sup> stage </b>into a desired altitude and <b>velocity</b> after which the
booster stage is way off the <b>launchpad</b>. To get back to the launchpad it needs
to cancel out the energy gained and regain the same amount of energy and it is
almost impossible considering the physics of our planet. The booster can never
take enough fuel with it to boost back to the launch site. So instead, <b>SpaceX</b>
planned for landing on a drone ship in the middle of the sea. But why and how
is boost back done anyways? Boost back is done just to <b>cancel out the excess
energy</b> before it re-enters the atmosphere with about <b>7 times the speed of
sound</b>. Now, here kicks in the entry burn. Without an entry burn, the rocket
will be torn apart mid-air. This burn slows down the rocket for a safe entry
into the atmosphere.</span><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"><o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Suicide Burn:<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjkhkevsVppP1HW8VhaVyEIS9ayKvLbJQ5zUxSIAtMjkZuBD-FiB2kKsgAVWmQpusWz_erQeFLUQHIVD0phqSrr1s0xoDeg8NDXHBbH7DVl3vFAbTKhnagPIw04ra0_OJoMFU3hA48MzJCbpzZD-ZQhhyIN0EqIYELzzPRlBZCDisT4W6o0jwC8Ma75=s1592" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="1592" data-original-width="1559" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjkhkevsVppP1HW8VhaVyEIS9ayKvLbJQ5zUxSIAtMjkZuBD-FiB2kKsgAVWmQpusWz_erQeFLUQHIVD0phqSrr1s0xoDeg8NDXHBbH7DVl3vFAbTKhnagPIw04ra0_OJoMFU3hA48MzJCbpzZD-ZQhhyIN0EqIYELzzPRlBZCDisT4W6o0jwC8Ma75=w313-h320" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="313" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Suicide burn</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Suicide Burn is a legit thing in <b>Kerbal Space
Program</b>. But apparently, SpaceX doesn’t like it to be called suicide burn. They
call it Hover Slam instead, so we’ll also call it <b>Hover slam</b> </span><span face=""Segoe UI Emoji",sans-serif" style="color: #0b5394; font-size: 13.5pt; mso-ascii-font-family: "Footlight MT Light"; mso-bidi-font-family: "Times New Roman"; mso-char-type: symbol-ext; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-hansi-font-family: "Footlight MT Light"; mso-symbol-font-family: "Segoe UI Emoji";">😉</span><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">
(just kidding). Suicide burn is like parking your car like the protagonist in <b>Rohit
Shetty</b> <b>films</b> but here it's with rockets. The rocket just heads to the landing site full
speed and at a certain point of time before touchdown, the middle <b><a href="https://www.thedynamicfrequency.org/2022/01/spaceX-merlin-intro-to-turbopump.html" target="_blank">merlin engine</a></b>
is turned on which slows the booster to almost <b>0 km/h </b>just before touchdown.
Now there is a reason why it is called suicide burn, it is because if you miss
the right time the booster ain’t going to land. If you <b>burn early</b>, you use up
the fuel and smash on the ground, if you <b>burn late</b> the booster still crashes.
Risky stuff indeed!!</span><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"><o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Soft Touchdown:<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiMVPLIdZaxYbnzgR4MaXA9KCht6BI-yLRANE9qwuo-9C5IzMP7Uta-wO3Pu_crnrZBtKA8qrM5XCL4qV7YDEgcGYMJB1WyQdIwYuKFTOh9B3MNyC8Kch--2bDRpN8s4cvwtsjgsZq-EyIqlYMqW49_XIKQIEm363O2Cp4PGOdk_isL_TB0w7zoJn55=s640" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="427" data-original-width="640" height="214" src="https://blogger.googleusercontent.com/img/a/AVvXsEiMVPLIdZaxYbnzgR4MaXA9KCht6BI-yLRANE9qwuo-9C5IzMP7Uta-wO3Pu_crnrZBtKA8qrM5XCL4qV7YDEgcGYMJB1WyQdIwYuKFTOh9B3MNyC8Kch--2bDRpN8s4cvwtsjgsZq-EyIqlYMqW49_XIKQIEm363O2Cp4PGOdk_isL_TB0w7zoJn55=w320-h214" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Iconic touchdown click from the drone ship</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The booster nailing the suicide burn ain’t making no
difference with defective landing legs. The Falcon9 uses <b>4
hydraulics-controlled legs</b>.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The legs have 4 major components;<o:p></o:p></span></p><p class="MsoListParagraphCxSpFirst" style="background: white; line-height: normal; margin-bottom: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; text-align: justify; text-indent: -18pt; vertical-align: baseline;"><!--[if !supportLists]--><span style="color: #0b5394; font-family: Symbol; font-size: 13.5pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: EN-IN;">·<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span></span><!--[endif]--><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The deployment arms<o:p></o:p></span></p><p class="MsoListParagraphCxSpMiddle" style="background: white; line-height: normal; margin-bottom: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; text-align: justify; text-indent: -18pt; vertical-align: baseline;"><!--[if !supportLists]--><span style="color: #0b5394; font-family: Symbol; font-size: 13.5pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: EN-IN;">·<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span></span><!--[endif]--><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The legs<o:p></o:p></span></p><p class="MsoListParagraphCxSpMiddle" style="background: white; line-height: normal; margin-bottom: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; text-align: justify; text-indent: -18pt; vertical-align: baseline;"><!--[if !supportLists]--><span style="color: #0b5394; font-family: Symbol; font-size: 13.5pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: EN-IN;">·<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span></span><!--[endif]--><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The pusher<o:p></o:p></span></p><p class="MsoListParagraphCxSpLast" style="background: white; line-height: normal; margin-bottom: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; mso-margin-top-alt: auto; text-align: justify; text-indent: -18pt; vertical-align: baseline;"><!--[if !supportLists]--><span style="color: #0b5394; font-family: Symbol; font-size: 13.5pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; mso-fareast-language: EN-IN;">·<span style="font-family: "Times New Roman"; font-size: 7pt; font-stretch: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;">
</span></span><!--[endif]--><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The latches<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The deployment arms: </span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg6PMd0JcNfmNyVTI0IdxRS5IjMPHg8GR9L-AR48LOKOPTz85OFh03lDTe9QBBq_sQDUjOGZGl5lV8zhmDn-p_QaCQQrzICewtXS_1_LsL0kbyFAFNn_kABqR2jwGhAEtDPfReddqmKNIUqONqFhPyJgVRmiG_NGZUsx1GdbXIW_WPsgq54fZM5rB9C" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" data-original-height="641" data-original-width="598" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEg6PMd0JcNfmNyVTI0IdxRS5IjMPHg8GR9L-AR48LOKOPTz85OFh03lDTe9QBBq_sQDUjOGZGl5lV8zhmDn-p_QaCQQrzICewtXS_1_LsL0kbyFAFNn_kABqR2jwGhAEtDPfReddqmKNIUqONqFhPyJgVRmiG_NGZUsx1GdbXIW_WPsgq54fZM5rB9C=w299-h320" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="299" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Telescopic deployment arms</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">If you have seen a <b>telescope</b>, then you are probably
aware of how it extends forward and retracts in, well, that design is called
<b>telescopic mechanism</b>. The deployment arms have the same mechanism to deploy the
<b>landing legs</b>. Normally, the arms are collapsed, when the booster nails the
suicide burn, then only deployment arms extend as <b>pressurized Helium</b> fills up
the arm. These arms also work as shock absorbers as there’s a heavy chance of
the <b>booster bouncing </b>back after the touchdown.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The legs:</span></b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiAYlC08EEXnyxCvk0iQCDpdfVW94VXtn1Vy5Tng4elA2GpnEePHvlii-ES0Dyxh8H4qsJZi4kyScSHEIL7cK08mni5aDH0iGh1SidPgN-WWBPQfsC6_qI4rLEhdNTpnXtFVY7bFy1SG56aFMpvvozuCR44d9Bah2jFUh0RTOggdvHtY9zlIr9J5DIO" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" data-original-height="316" data-original-width="1186" height="85" src="https://blogger.googleusercontent.com/img/a/AVvXsEiAYlC08EEXnyxCvk0iQCDpdfVW94VXtn1Vy5Tng4elA2GpnEePHvlii-ES0Dyxh8H4qsJZi4kyScSHEIL7cK08mni5aDH0iGh1SidPgN-WWBPQfsC6_qI4rLEhdNTpnXtFVY7bFy1SG56aFMpvvozuCR44d9Bah2jFUh0RTOggdvHtY9zlIr9J5DIO=w320-h85" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Legs</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Probably the <b>sexiest legs </b>ever built, these are the parts that physically touch the ground, they have <b>grooves </b>to
house the deployment arms. The legs are normally closed and open with the deployment
of the deployment arms.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The pushers: </span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjCfN5xHI15AaSoEIovaSLXG9I0oR-D35_ZsAIAb2h6C5ZW55x3D-Tn_7diRI6gtpEJCDmEGI_f7Ttp7F0RtsmC5sk2RFEId0dGd40Det7Q5UMs88ikTdZafcvkV_aB9Bo07FzNZnpqALWZ7ZngO7TgokaT3y7GL9Wn09XCbe9ohNQaLpkMT9W9Bw8b" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" data-original-height="79" data-original-width="93" height="271" src="https://blogger.googleusercontent.com/img/a/AVvXsEjCfN5xHI15AaSoEIovaSLXG9I0oR-D35_ZsAIAb2h6C5ZW55x3D-Tn_7diRI6gtpEJCDmEGI_f7Ttp7F0RtsmC5sk2RFEId0dGd40Det7Q5UMs88ikTdZafcvkV_aB9Bo07FzNZnpqALWZ7ZngO7TgokaT3y7GL9Wn09XCbe9ohNQaLpkMT9W9Bw8b=w320-h271" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Zoomed-in pic of pushers</td></tr></tbody></table><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><br /><span style="font-size: 13.5pt;">These
components are too small and barely visible to us. As you can see in the images
that I’ve attached here, the pushers look like small antenna-like structures. Their
main purpose of existence is to <b>kick the legs slightly off the body </b>so that gravity
and the main arms can do their respective jobs.</span></span><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><b>The latches:</b> </span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjUCo1_fMzxDwBi0x6IU6OwVosVe-iKK7cxG0_3Rv2WzDt4fh8A6I0yh8bVoMYYro7zvhJDUVfzyskGq2SrUK7dYXJnZw29vfteO6Yt7-olQp7N1EuazC3qP0pcUINMlZKCOimnKetsGCjed2MVcgixI7Xg8Q93PSFSu8_nrP9nyMn_jm1vRgw-JV3Q=s3000" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="2000" data-original-width="3000" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEjUCo1_fMzxDwBi0x6IU6OwVosVe-iKK7cxG0_3Rv2WzDt4fh8A6I0yh8bVoMYYro7zvhJDUVfzyskGq2SrUK7dYXJnZw29vfteO6Yt7-olQp7N1EuazC3qP0pcUINMlZKCOimnKetsGCjed2MVcgixI7Xg8Q93PSFSu8_nrP9nyMn_jm1vRgw-JV3Q=w320-h213" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Visible triangular latches</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">As the name suggests, these small
triangular objects stop the legs from deploying during ascent. These open if
and only if the<b> booster does a suicide burn</b>.</span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEghxdDzoJFhf86KZ5yhRjrUkGNhCqXJWtn6JciNwpSH3wud874_wTbWth9qh6avwJedyNb5YZATiB1U5hPWzk0Cha8LwyXVoA1P8FyPny57Enc7hGy6HrY01RrgDMPG2DvitbazISPK5vavFFTkI3V-kmgjoaP6gynlBZuPq78sazz_kJ1vxcwj0Iug=s2633" style="margin-left: auto; margin-right: auto;"><img alt="How to Land an Orbital Class Rocket...The Dynamic Frequency" border="0" data-original-height="1499" data-original-width="2633" height="182" src="https://blogger.googleusercontent.com/img/a/AVvXsEghxdDzoJFhf86KZ5yhRjrUkGNhCqXJWtn6JciNwpSH3wud874_wTbWth9qh6avwJedyNb5YZATiB1U5hPWzk0Cha8LwyXVoA1P8FyPny57Enc7hGy6HrY01RrgDMPG2DvitbazISPK5vavFFTkI3V-kmgjoaP6gynlBZuPq78sazz_kJ1vxcwj0Iug=w320-h182" title="How to Land an Orbital Class Rocket...The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The Falcon 9 trajectory</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">These 4 major components work <b>hand in hand</b> to make a
near-perfect landing possible. It’s surprising the Falcon-9 booster can land
with a <b>landing accuracy of 10m</b> that’s bonkers considering the booster makes a
comeback from near orbital velocity. <b>Insane engineering spotted.</b><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">If
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on mobile).</b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: #474747; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">To know the basics of the quantum world, astronomy
and space exploration you can check out the book </span></b><a href="https://www.blogger.com/blog/post/edit/3578987301814134971/9084617100284295853"><b><span style="border: 1pt none windowtext; color: #e94bc3; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">"Through the wormhole" on amazon kindle</span></b></a><b><span style="border: 1pt none windowtext; color: #474747; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"> by Ratnadeep Das Choudhury.</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; 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border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></span></div></div></div><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; vertical-align: baseline;"><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 11.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; vertical-align: baseline;"><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p align="right" class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: right; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">
</span><span style="color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">-Ayushman Dash</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p align="right" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">
Chief Space Exploration
Writer</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p align="right" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
Dynamic Frequency</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p class="MsoNormal"><o:p> </o:p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;">
</p><p class="MsoNormal"><o:p> </o:p></p>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com2tag:blogger.com,1999:blog-3578987301814134971.post-88680487150642391642022-02-23T20:33:00.009+05:302022-02-23T20:39:00.930+05:30Quantum Entanglement: Bohr-Einstein debates<p></p><p class="Standard"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiLxtZbuAb13sl7IkY4tP_FjymUoND3JbENP1H5TlT-iQV96HOKndWJC3J8oqX9_pKO8ZVhCkA6-WR0X7qB_uJDnjSUj1_eCrVNQ7GFU-RA1FFM5pA6BYsKx3zuIze1Phj-hlF6f7jp25-B6bu3cEXjMRapX31DjHhADaAus1-UtaPF-5MqWqEJLfVU3Q=s647"><span style="color: #0b5394;"><img alt="Quantum Entanglement - Bohr-Einstein debates: the dynamic frequency" border="0" data-original-height="340" data-original-width="647" height="168" src="https://blogger.googleusercontent.com/img/a/AVvXsEiLxtZbuAb13sl7IkY4tP_FjymUoND3JbENP1H5TlT-iQV96HOKndWJC3J8oqX9_pKO8ZVhCkA6-WR0X7qB_uJDnjSUj1_eCrVNQ7GFU-RA1FFM5pA6BYsKx3zuIze1Phj-hlF6f7jp25-B6bu3cEXjMRapX31DjHhADaAus1-UtaPF-5MqWqEJLfVU3Q=w320-h168" title="Quantum Entanglement - Bohr-Einstein debates: the dynamic frequency" width="320" /></span></a></div><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><br /> </span><p></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p></o:p></span></p><blockquote><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;">“The
most bizarre, the most absurd, the most crazy prediction that quantum mechanics
makes is, Entanglement.”</span></span></blockquote><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span><p></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">~
Walter Lewin, 2014</span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> </span></p>
<p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">Everyone
of us taking some interest in quantum mechanics, must have heard of this
perplexive phenomenon, </span><b><span style="color: #0b5394;">“Entanglement”</span><span style="color: #0b5394;">.</span></b><span style="color: #0b5394;"> What is it simply? Does it really
happen? How does it help? Let’s dive right in to find out! Do checkout our </span><b><span><a href="https://www.thedynamicfrequency.org/2019/09/introduction-to-quantumEntanglement.html" target="_blank">previous introductory article</a></span></b><span style="color: #0b5394;"> on quantum entanglement for more learning!</span></span></p><p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></p>
<p class="Standard" style="text-align: left;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light",serif; font-size: x-large;">What
is Entanglement?<o:p></o:p></span></b></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Entanglement
is a theoretical prediction that comes from the equation of quantum mechanics!
<b>Any two particles can become entangled if they are close together and then
their properties become linked</b>. That is, any actions on one particle would not
only <b>affect</b> that particle, but also <b>counter-affect</b> it’s entangled partner no
matter how far they are! This makes sense when the particles are placed very
close. But remarkably, quantum mechanics says that even if you separated those
particles, sending them in opposite directions, they would still remain
entangled, no matter how much the distance is between them and whether or not
there is a direct medium connecting both the particles! How weird is it? It’s
like you turn on your bluetooth in India and your brother’s airpods disconnects
from his phone in America! (<b>action and counter-action</b>)<o:p></o:p></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> </span></p>
<p class="Standard" style="text-align: left;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light",serif; font-size: x-large;">Two
school of thoughts for Entanglement<o:p></o:p></span></b></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Niels
Bohr and Albert Einstein were two among the many significant scientists
researching on quantum mechanics. While Bohr was confident in the mathematics
of quantum mechanics and thus, laid out all the results of it, Einstein always
believed that quantum mechanics lacked something that makes it incompetent to
define the reality, unlike classical physics does. Let’s take a look at both of
their perceptions of quantum mechanics.<o:p></o:p></span></p><p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><br /></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p></o:p></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhabvG50Bihaept-jbvzy18MVA_7m4TcVKpno0b7ThsNHMPl8xaNQBuLkyc8zhVMdHKooYswsc1QyYNP7SVMJBYjTy_t4vTxxEirXCXfrI710WADX7VSs69iPg5Dno-5BvgRnOaf24NUoYEr_Gdrg2FiBRpH6fJ8hSOf8p-bHJB58nKVUfg0h0ybha78g=s700" style="margin-left: auto; margin-right: auto;"><span style="color: #0b5394;"><img alt="Bohr-Einstein debates : the dynamic frequency" border="0" data-original-height="435" data-original-width="700" height="199" src="https://blogger.googleusercontent.com/img/a/AVvXsEhabvG50Bihaept-jbvzy18MVA_7m4TcVKpno0b7ThsNHMPl8xaNQBuLkyc8zhVMdHKooYswsc1QyYNP7SVMJBYjTy_t4vTxxEirXCXfrI710WADX7VSs69iPg5Dno-5BvgRnOaf24NUoYEr_Gdrg2FiBRpH6fJ8hSOf8p-bHJB58nKVUfg0h0ybha78g=w320-h199" title="Bohr-Einstein debates : the dynamic frequency" width="320" /></span></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #e94bc3;">Bohr - Einstein debates</span></td></tr></tbody></table><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><br /> </span><p></p>
<p class="Standard"><b><span lang="EN-US" style="color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Niels
Bohr<o:p></o:p></span></b></p>
<p class="Standard"><b><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">Observation
implies distinct action</span><span style="color: #0b5394;">,</span> </span></b><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">is what Bohr emphasized on since the
beginning. We have already talked about the role of </span><b><span style="color: #0b5394;">“observation”</span> </b><span style="color: #0b5394;">in
quantum mechanics. Any event or particle is a distribution of probabilities in
the quantum world. And <b>only when an observation attempt is made on the
particle/event, it acquires a particular state</b>! So, entanglement states that if
you measure a particle here, not only would you affect it’s state (one out of
the thousand possibilities), but your measurement would also affect the state
of it’s entangled partner no matter how distant they are! There is no force, or
no wires or anything connecting them! There’s no way they could communicate
with each other. Sounds so unacceptable right? More like a philosophy rather
than a science! But this is what <b>Bohr excellently derived from the equations
and maths of quantum mechanics!</b><o:p></o:p></span></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></p>
<p class="Standard"><b><span lang="EN-US" style="color: #04ff00; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Albert
Einstein<o:p></o:p></span></b></p>
<p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">This
kind of weird long-range connection between particles sounded so absurd to
Einstein that he called it spooky, </span><b><span style="color: #0b5394;">spooky action at a distance!</span></b><span style="color: #0b5394;"> Amidst
these situations, Einstein suggested that Entanglement would be possible, but
not the way this is described as a </span><span style="color: #0b5394;"><b>spooky action at a distance.</b></span><span style="color: #0b5394;"> Let’s
take an example to understand the ideology of Niels Bohr and Albert Einstein.<o:p></o:p></span></span></p>
<p class="Standard"><b><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></b></p>
<p class="Standard"><b><span lang="EN-US" style="color: #04ff00; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Examples<o:p></o:p></span></b></p>
<p class="Standard"><b><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></b></p>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiqU2XAojbNQlwDmTVwdKEJdQIxMQ5GTRkhD05HaD1fU03FzhFeuOKuAzyd4waFSTtqXBB04Z5ntI-Z8xKL3-lZDzcyzddLaN7MMZeopqhnyWKPmSTvgaGtdEiFBj0vKPqrqD1E_ZzuoyCIFByLYNh-_Q5bkqXccsby_ftbOrBF_eyeiEbag6Pz_uLy3g=s600" style="margin-left: auto; margin-right: auto;"><span style="color: #0b5394;"><img alt="Pair of entangled gloves : the dynamic frequency" border="0" data-original-height="600" data-original-width="600" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEiqU2XAojbNQlwDmTVwdKEJdQIxMQ5GTRkhD05HaD1fU03FzhFeuOKuAzyd4waFSTtqXBB04Z5ntI-Z8xKL3-lZDzcyzddLaN7MMZeopqhnyWKPmSTvgaGtdEiFBj0vKPqrqD1E_ZzuoyCIFByLYNh-_Q5bkqXccsby_ftbOrBF_eyeiEbag6Pz_uLy3g=w320-h320" title="Pair of entangled gloves : the dynamic frequency" width="320" /></span></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #e94bc3;">A pair of entangled gloves</span></td></tr></tbody></table><span style="color: #0b5394;"><br /></span><p class="Standard"><b><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></b></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><b>Entangled
particles can be like a pair of gloves.</b> Imagine someone separates the two gloves
and puts them one in each box. Then, he delivers one of the box to me, in India
and the other to you, in Antarctica (well you would be a penguin that way then
xD). So before I looked inside my box, I know it has either a left hand glove
or a right hand glove. And the moment when I open my box, let’s say I found a
left hand glove, then at that particular instant, I’m sure that the box in
Antarctica would contain a right hand glove, even though you have not looked
inside it yet!</span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">There
is nothing mysterious about this, says <b>Einstein</b>, because <b>obviously by looking
inside the box, I haven’t affected either of the glove boxes</b>. My box always had
a left hand glove and thus the box in Antarctica always had a right hand glove,
since the moment they were separated! But <b>Bohr</b> would believe that the <b>glove in
my box was equally happening to be a left handed glove and a right handed
glove.</b> And only in the instant where I took the observation by opening the box,
the glove resolved into a specific configuration. And thus, this affected the
other glove box in Antarctica, making it to be it’s opposite configuration glove
anyhow.</span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> </span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhl_SkLv_HARPBxAry9bHnV2aew9o0tqsx1fD1Ufu4YLBYjK1HovKcxJaV49Pj9m21e4o6lNXJap-CdBt8wTMKR7eX2pqzvZR0fmeNe3uZ7AshF8eaHhvD3LIBQASNN5EISHuNSqtaUhF_kkFZ2mx_IKBiEKPjP4fmtuhiRXDR_ZH_kIUkyAFRbbqtApA=s510" style="margin-left: auto; margin-right: auto;"><span style="color: #0b5394;"><img alt="Electron spin : the dynamic frequency" border="0" data-original-height="340" data-original-width="510" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEhl_SkLv_HARPBxAry9bHnV2aew9o0tqsx1fD1Ufu4YLBYjK1HovKcxJaV49Pj9m21e4o6lNXJap-CdBt8wTMKR7eX2pqzvZR0fmeNe3uZ7AshF8eaHhvD3LIBQASNN5EISHuNSqtaUhF_kkFZ2mx_IKBiEKPjP4fmtuhiRXDR_ZH_kIUkyAFRbbqtApA=w320-h213" title="Electron spin : the dynamic frequency" width="320" /></span></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #e94bc3;">Electron Spin</span></td></tr></tbody></table><p></p><p class="Standard"><span style="color: #0b5394;"><br /></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><b>Einstein</b>
believed this exactly applied to particles as well. <b>Whatever configuration the
separated entangled particles are in, must have been fully determined from the
moment they flew apart!</b> Einstein and Bohr would often make their contradicting
points regarding Entanglement! Electron can have two spins right; clockwise and
counter-clockwise.</span></p><p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Imagine two entangled electrons are separated away very far.
And then the spin of one of them is measured to be clockwise. At this instant,
the spin of the other electron is bound to be counter-clockwise. So Einstein
would say that this electron has a definite spin before you measure it. Bohr
would ask how would you check that? Einstein says, measure it! You will find
the definite spin! Bohr would nod, but its the act of measurement that brought
the spin of the electron to a definite state. Boom, mindblowing right?<o:p></o:p></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> </span></p>
<p class="Standard" style="text-align: left;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light",serif; font-size: x-large;">Who
was right?<o:p></o:p></span></b></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><br /></span></p><p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">So,
the important question, </span><b><span style="color: #0b5394;">who between Bohr and Einstein was right regarding
Entanglement?</span></b></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Niels
Bohr and Albert Einstein have both spent a good amount of time in researching
quantum mechanics. They were two different school of thoughts. Einstein always
believed that quantum mechanics is lacking something which doesn’t make it
acceptable, and even persisted this thought till his demise in 1955!</span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> </span></p>
<p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">In
1967, John Clauser, a PhD student as a part for his academics, carried out an
experiment to prove Entanglement and hence, the whole quantum mechanics <b>wrong!</b>
He devised a machine that could measure thousands of pairs of entangled
particles and compare their spins in different directions. Finally something to
settle the debate between Bohr and Einstein! As the result started to come in,
Clauser was surprised! <b>All of the thousand pairs of entangled particles
followed the rule of entanglement!</b> Measuring spin of one of the particles in
the device instantly affected the spin of the other particle! Thus, </span><span style="color: #0b5394;"><b>Clauser
in an attempt to rule out quantum mechanics, proved that all the maths of
quantum mechanics is correct! Entanglement is real!</b></span></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">So,
Niels Bohr, who championed the equations, was absolutely right with all of his
mathematics about quantum mechanics!</span><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"> <b><span style="color: #0b5394;">Entanglement, is real.</span></b></span><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"> The one
thing that Einstein thought was impossible, spooky action at a distance,
actually happens :O<o:p></o:p></span></p>
<p class="Standard"><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><o:p> </o:p></span></p>
<p class="Standard" style="text-align: left;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light",serif; font-size: x-large;">How
does Entanglement help?</span></b></p><p class="Standard"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt;"><span style="color: #0b5394;">When
was the last time you saw teleportation in movies? Ever thought if it was
possible in real life? Well, Entanglement! </span><span style="color: #0b5394;"><b>Entanglement promises some
progress in quantum teleportation.</b></span><span style="color: #e94bc3;"> </span></span><span lang="EN-US" style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt;">Apart from that, it also supports many
fields like quantum computing, quantum cryptography, quantum dense coding etc.
I would definitely take all of these interesting topics next, and share my thoughts
on them! Stay tuned for the next one, hope you liked it :)</span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">If
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and space exploration you can check out the book </span></b><a href="https://www.amazon.in/dp/B07TBSP6JT"><b><span style="border: 1pt none windowtext; color: #e94bc3; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">"Through the wormhole" on amazon kindle</span></b></a><b><span style="border: 1pt none windowtext; color: #474747; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"> by Ratnadeep Das Choudhury.</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 11.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><br />
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</span><span style="color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">-Beleswar Prasad Padhi</span></p><p align="right" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">
Chief Quantum-verse Writer</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p><p align="right" class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
Dynamic Frequency</span></p><p></p>Beleswar Prasad Padhihttp://www.blogger.com/profile/01830716348127057293noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-9404699805755543602022-02-05T21:54:00.007+05:302022-03-09T21:39:46.499+05:30Saturn-V for Dummies Part-5: Conclusion<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiu5y_OiNCcyWQ1EQKDeeEKS2FG3l6xLgby8eN7-jL3Xdq2-He94A2cVE4zbu9es1Lpmtbf6UjsBUK8NNGLSMms3karPx13RXVk0ePf_g3r1YI-HTg7lVqqDwg2rCCZoSobIbnnPUbxQgI8OWqaF7CUPawmHFSqo_jkpbkNK9BaUK5OlU9V7UlWCrJm=s640" style="margin-left: 1em; margin-right: 1em;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="514" data-original-width="640" height="257" src="https://blogger.googleusercontent.com/img/a/AVvXsEiu5y_OiNCcyWQ1EQKDeeEKS2FG3l6xLgby8eN7-jL3Xdq2-He94A2cVE4zbu9es1Lpmtbf6UjsBUK8NNGLSMms3karPx13RXVk0ePf_g3r1YI-HTg7lVqqDwg2rCCZoSobIbnnPUbxQgI8OWqaF7CUPawmHFSqo_jkpbkNK9BaUK5OlU9V7UlWCrJm=w320-h257" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></div><p></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><b></b></span></p><blockquote><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><b>A conclusion is simply the
place where you get tired of thinking</b></span><b> </b></blockquote><p></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">~
Dan Chaon, Stay Awake<o:p></o:p></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Hello
all, <b>Happy Basant Panchami</b>, and welcome to the final blog of this series, where I will try to answer some
frequently asked questions about the mighty Saturn-V and the moon missions in
general. The previous 4 articles were about <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt1-history.html" target="_blank">the history</a></b>, <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturnV-for-dummies-pt2-machine-overview.html" target="_blank">engineering</a></b>,
<b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt-3-the-engines.html" target="_blank">engines</a></b>, and <b><a href="https://thedynamicfrequency.blogspot.com/2022/02/saturn-v-for-dummies-pt4-missions.html" target="_blank">missions</a> </b>of the rocket (highly recommended going
through them, please give them a read). I tried to cover the rocket from every
possible angle in all four with a generalized approach. But today I shall be
concluding this series, but is it the end? Watch me answer this question and
more as you read and scroll.<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><b><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 24pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Conclusion:
Or is it the end?</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Let’s
start with the questions:<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: What happened to the Saturn-V?<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiYjdVbkDwJ54C9MlQQ8kMpJyYNPIodjd8HKgZrhGqR3YOvUXskocA9n5d-Pkt18SryJOnvbQlBEubob9AxELlAxICONSgNosSyBGD38zMYoBqRCbfPUCRSYdmfq-QkvJDslyC8drwHI7PWwCm4H_NpRC-xC9Y4na5CoN34f7Ym-pZ271w92lB8to5H=s640" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="440" data-original-width="640" height="220" src="https://blogger.googleusercontent.com/img/a/AVvXsEiYjdVbkDwJ54C9MlQQ8kMpJyYNPIodjd8HKgZrhGqR3YOvUXskocA9n5d-Pkt18SryJOnvbQlBEubob9AxELlAxICONSgNosSyBGD38zMYoBqRCbfPUCRSYdmfq-QkvJDslyC8drwHI7PWwCm4H_NpRC-xC9Y4na5CoN34f7Ym-pZ271w92lB8to5H=w320-h220" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Apollo module docking with the Skylab</td></tr></tbody></table><p></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A: Saturn-V has not flown since <b>May 14, 1973</b>, as it
retired after the last <b>Skylab mission</b> which I covered in detail in the "<b><a href="https://thedynamicfrequency.blogspot.com/2022/02/saturn-v-for-dummies-pt4-missions.html" target="_blank">Missions</a>"</b>
blog. Make sure to check it out for more information.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: Why did the Saturn-V retire?<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><div class="separator" style="clear: both; text-align: center;"><b><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgYArnaE7Za0YZG6zfVV_JmaZjr62_WHiXcP-PB9SrJwXTryj_mH-DFCgoUufRcLVhZk7B9ZFfeRDgwZMgoqMnM2l_fj2r7Dl2Cy7cCLUQm0X04-ivNqBiBVnw0Hnbthl_ghm6t8brlWIm7jnkrAB0ysytxPHnA8BcfuakLKHd0r4F6UInMVNN-sV1S=s1920" style="margin-left: 1em; margin-right: 1em;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="1335" data-original-width="1920" height="223" src="https://blogger.googleusercontent.com/img/a/AVvXsEgYArnaE7Za0YZG6zfVV_JmaZjr62_WHiXcP-PB9SrJwXTryj_mH-DFCgoUufRcLVhZk7B9ZFfeRDgwZMgoqMnM2l_fj2r7Dl2Cy7cCLUQm0X04-ivNqBiBVnw0Hnbthl_ghm6t8brlWIm7jnkrAB0ysytxPHnA8BcfuakLKHd0r4F6UInMVNN-sV1S=w320-h223" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></b></div><p></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A: There are several valid reasons for the <b>US govt</b>
and <b>NASA</b> to retire the Saturn-V rocket. Some of them are; cost. It cost around
a <b>47billion USD</b> in <b>10 years</b> for <b>13 launches</b>, which boils down to around <b>3.6
billion USD per launch</b>, and from this figure, if you subtract the developmental
cost, it will cost merely <b>1.2 billion USD per launch</b>. Another reason is pay
cut. The govt. decided to restrict the budget of NASA which resulted in
stopping further production of the rocket and the rest got distributed among <b>space
museums for display</b> and study for the common people. NASA with the restricted
budget shifted its focus to the <b>Space Shuttle Program</b>.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: Was moon landing fake?<o:p></o:p></span></b></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgJezRuv8cND4HYvSqvcnJhKgDk2uIthTdM7qaalGZUW00E29iCXcjHkzGQcaiu94pmO5MJqcT2mSnCLwrlzDaUUMmfVoqvNB7rhhNCJWG0trra71IUdBdzcPlLPqDJqabjrxprKVkeUXiaAWIvAzBlMgMGw_FVrfcNZZIKbatVxEkag2YilZ8HB_6Y=s1280" style="margin-left: 1em; margin-right: 1em;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="1188" data-original-width="1280" height="297" src="https://blogger.googleusercontent.com/img/a/AVvXsEgJezRuv8cND4HYvSqvcnJhKgDk2uIthTdM7qaalGZUW00E29iCXcjHkzGQcaiu94pmO5MJqcT2mSnCLwrlzDaUUMmfVoqvNB7rhhNCJWG0trra71IUdBdzcPlLPqDJqabjrxprKVkeUXiaAWIvAzBlMgMGw_FVrfcNZZIKbatVxEkag2YilZ8HB_6Y=w320-h297" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></div><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A: There has been a particular school of people who
believe the moon landing was a fake act under the direction of NASA. Now look,
normally I am not an advocate of violence but I believe these <b>conspiracy
theorists</b> should be packed inside a <b>mental asylum and beaten</b>. These conspiracy
theorists state several reasons and excuses for their belief, which are insane and baseless.
If you need to prove it fake, then you need to prove that the whole world was
lying about the moon landing as every<b> space-faring country</b> recorded the moon
landing and had it <b>confirmed</b>. It’s utter disrespect to all those scientists,
engineers, and several other people who made the moon mission possible. The
straight answer is <b>NO</b>, the moon landing was <b>NOT FAKE</b>.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: Why haven’t humans returned to the moon since
1972?<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi58HrxMO2ZDnplh9yDtv2NvGBA10L5MKM10GquH9jWuy8Vce78ZISzTMx94-OAo3mGT9R4YZKgPIs2umrmJX2FAhzBGulk4FC3ANRkiOMiUq15Jg1Ra92C8Q_xzJD58_lF2-iOx_rlItjk0psdoK2rbIsBtbmPKogR09zGF3cURQR8VQF1I4JN7fdJ=s400" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="400" data-original-width="100" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEi58HrxMO2ZDnplh9yDtv2NvGBA10L5MKM10GquH9jWuy8Vce78ZISzTMx94-OAo3mGT9R4YZKgPIs2umrmJX2FAhzBGulk4FC3ANRkiOMiUq15Jg1Ra92C8Q_xzJD58_lF2-iOx_rlItjk0psdoK2rbIsBtbmPKogR09zGF3cURQR8VQF1I4JN7fdJ=w80-h320" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="80" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">China's Long March 9</td></tr></tbody></table><p></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A: This specific question can be divided into <b>2
separate</b> questions; first, <b>why hasn’t any American returned to the moon?</b> And
second, <b>why hasn’t there been any other country to launch humans to the moon?</b>
The simple and generalized answer to both the questions is money and power. In
the <b>1960s and 70s </b>the <b>cold war</b> was going on, only <b>2 nations</b> had such massive
capital and engineering know-how to pull out this massive task of landing humans
on the moon. So, this became a war whose outcome was crystal clear as<b> Sir Neil
Armstrong</b> set his first foot on the moon. With the collapse of the <b>Soviet
Union</b>, the <b>US</b> had no one to fight with and compete against, which led to a <b>cut
in NASA budget</b>. But today, NASA is determined to land the<b> first man of color
</b>and the <b>first woman </b>on the moon by <b>2023</b> with their massive <b>SLS rocket</b>. Talking
about other countries, very few countries have a space research program, out of
which if we deduct the <b>US</b>, <b>Russia</b>, and <b>the EU,</b> we are left with <b>China </b>which has<b> a similar space budget </b>as that of <b>the States</b> and they
have plans for a manned moon flight with their <b>super-heavy launch vehicle</b>.
So, although it has been a disappointment not to have watched the moon landing
on TV back then, today, we are living in the <b>best days of the space era</b> and
will soon watch <b>humans land on the moon</b>.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: Why is NASA using SLS for the Artemis program instead
of Saturn-V which has been already tried and tested several times?<o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjhqX0a_02HTHXFs47Gd5jeIFKjCg_gxFqDgqL0TfrGnMUZMNNdiwr4jjVwW25IpTp17IlYjtfTcxbJQGmqVyH5DieujuAdbwr9FCcNreRSM_o5cj8bT39TZvzkIbOrp1MfTQLjDiH7kSA2OUrN0KwXkoqBdDw0ZUuWO7wjhFL7rPzk3HzHZYZ-KwEu=s2940" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="2214" data-original-width="2940" height="241" src="https://blogger.googleusercontent.com/img/a/AVvXsEjhqX0a_02HTHXFs47Gd5jeIFKjCg_gxFqDgqL0TfrGnMUZMNNdiwr4jjVwW25IpTp17IlYjtfTcxbJQGmqVyH5DieujuAdbwr9FCcNreRSM_o5cj8bT39TZvzkIbOrp1MfTQLjDiH7kSA2OUrN0KwXkoqBdDw0ZUuWO7wjhFL7rPzk3HzHZYZ-KwEu=w320-h241" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The SLS rocket with Orion Space Craft</td></tr></tbody></table><p></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A: Saturn-V is a<b> marvel</b> not because it was built in
<b>recent times</b>, but because it was built back in the <b>1960</b>s, whose technology is no
match with today’s technology. Today one can <b>3d print rocket engines</b> completely
and even companies like <b>Relativity</b> are <b>3d printing entire rockets from top to
bottom </b>within days. But still, with those outdated methods, they achieved the impossible
feat. Manufacturing a Saturn-V now will be<b> costly</b> first of all and the
technology is <b>obsolete</b> now. The present SLS costs roughly <b>one-eighth</b> of what
Saturn-V used to cost (<b>500 million USD</b>). Apart from that, we are living in an era of <b>reusability
</b>and <b>cost reduction</b>, where <b>SpaceX’s rocket-boosters</b> can <b>self-land</b>, be <b>refurbished</b>,
and <b>reused</b> again. Even the SLS feels obsolete against such rockets. So, it
makes sense to make a <b>better and cheaper rocket</b> to land humans on the moon rather
than building a whole new Saturn-V.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Q: Is it the end?<o:p></o:p></span></b></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">A:
<b>NO</b>, Saturn-V’s legacy continues. No matter where you live or what your identity
is, whether <b>American or non-American</b>, <b>white or black or colored</b>, <b>male or female</b>
you need to feel proud that humanity has <b>persevered</b> and reached the moon only within
<b>seven decades</b> after inventing the <b>first flying machine</b>. The dreams you think are
impossible to achieve now, just remember that landing on the moon used to be a
<b>dream</b> too but the dream saw the daylight with the help of sweat, tears, and
blood of the <b>innumerable</b>. <o:p></o:p></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh3u5DbV5Gq_ke7bkwzR2nXo901_XI1L0u53AmzGh5YrUGOhnumopAmowz9BOkw0hmDR8G5WwIJwy0iEFEPC7ZglCxlCX7Nkj31q7LJ5f4F0RGOz4Mb2oGG1f2ChsosQ6cUjeAcctcLES9TWGv6MxyLnshCTSvNfxWGUeohWIakH7wh46-1SkG9JwR_=s5516" style="margin-left: 1em; margin-right: 1em;"><img alt="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" border="0" data-original-height="3649" data-original-width="5516" height="212" src="https://blogger.googleusercontent.com/img/a/AVvXsEh3u5DbV5Gq_ke7bkwzR2nXo901_XI1L0u53AmzGh5YrUGOhnumopAmowz9BOkw0hmDR8G5WwIJwy0iEFEPC7ZglCxlCX7Nkj31q7LJ5f4F0RGOz4Mb2oGG1f2ChsosQ6cUjeAcctcLES9TWGv6MxyLnshCTSvNfxWGUeohWIakH7wh46-1SkG9JwR_=w320-h212" title="Saturn-V for Dummies Part-5: Conclusion The Dynamic Frequency" width="320" /></a></div><p></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">I
chose to explain the Saturn-V rocket because it teaches us that<b> nothing is
impossible </b>and <b>everything is achievable</b> if we truly persevere for it. The
legacy of this rocket<b> inspires</b> me more than any human that walked on earth. I
won’t stop talking about the rocket. Though the <b>future articles</b> will focus on
other topics, still there will be subtle references to the rocket. <b>There’s more
rocketry stuff pouring in with time, stay tuned</b>. Once the next article is out,
the link will be available <b>here</b>.<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">If you like our articles then please <b>follow this blog</b> to get the <b>latest updates directly to you</b> <b>(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><b style="background: 0px 0px; border: 0px; color: #474747; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">To know the basics of the quantum world, astronomy and space exploration you can </b><b style="background: 0px 0px; border: 0px; color: #474747; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">check out the book <a href="https://www.amazon.in/dp/B07TBSP6JT" style="background: 0px 0px; border: 0px; color: #e94bc3; outline: 0px; padding: 0px; text-decoration-line: none; transition: color 0.17s ease 0s; vertical-align: baseline;" target="_blank">"Through the wormhole" on amazon kindle</a> by Ratnadeep Das Choudhury.</b></p><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; 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padding: 0px; vertical-align: baseline;"><i style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></i></div></div><div style="background: 0px 0px; border: 0px; color: red; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;">See you Again!!!!!</span></div><div><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></span></div></div></div><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; vertical-align: baseline;"><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p align="right" class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"> </span><span style="background: 0px 0px transparent; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background-color: transparent; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt;">-Ayushman Dash</span></p><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; font-family: "times new roman"; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"> Chief Space Exploration Writer</span></div><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">The Dynamic Frequency</span></div></div>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-89315806430840277702022-02-01T22:20:00.018+05:302022-02-23T00:05:55.535+05:30Saturn-V for Dummies Part-4: Missions <p><span style="color: #0b5394; font-family: "Footlight MT Light", serif; text-align: justify;"><span style="font-size: large;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgkfItEfkz9otnL2dfzeh_apa12V0PTvOiNGnaC1HuCqQD0rnKnyOS8xXBZf0_1BtYxhbbOxmdIxJA115Mh--oVGfovbNb4ImBP_S-EOIUGDulZcMLxQydZvSNFcsMUkPuL9hnCjG53r3E7NS26W1miXJAmhKfj9nLo70zHXj2Hky8G0rcLt9fHMMnC=s1024" style="margin-left: 1em; margin-right: 1em;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="610" data-original-width="1024" height="191" src="https://blogger.googleusercontent.com/img/a/AVvXsEgkfItEfkz9otnL2dfzeh_apa12V0PTvOiNGnaC1HuCqQD0rnKnyOS8xXBZf0_1BtYxhbbOxmdIxJA115Mh--oVGfovbNb4ImBP_S-EOIUGDulZcMLxQydZvSNFcsMUkPuL9hnCjG53r3E7NS26W1miXJAmhKfj9nLo70zHXj2Hky8G0rcLt9fHMMnC=w320-h191" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><b style="font-size: x-large;"></b></span></p><blockquote><span style="font-size: medium;"><b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">T</span><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">hat's one small step for man. One giant leap for mankind.</span></b></span></blockquote><span style="font-size: medium;"><b style="color: #0b5394; font-family: "Footlight MT Light", serif;"></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">~ Neil Armstrong</span></span><p></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><span style="font-size: medium;">H</span></span><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">ello
readers and welcome back, it has been quite a journey exploring the Saturn-V. In
our path, we came across 3 major milestones; 1<sup>st</sup> one was the <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt1-history.html" target="_blank">history of the development</a> </b>of the rocket, then we got an <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturnV-for-dummies-pt2-machine-overview.html">overview of the whole machine</a></b>, then we got to know the <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt-3-the-engines.html" target="_blank">propulsion system or the engines</a></b> of the rocket, if you haven’t read them, you are missing out on a tonne, links are
provided go and check it out first.<o:p></o:p></span></p></div>
<p class="MsoNormal" style="line-height: normal; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Hope
you are back after reading all of them. So, welcome back again!!<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Today,
we are going to see the missions that the vehicle accomplished. There was a
total of <b>13 launches</b> of the beast out of which <b>12 were a success</b> and <b>1
(Apollo-6) was a partial failure</b>. But, was Saturn-V only used for moon
programs? Let’s dive in and answer the question.<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><b><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 24pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Missions:
Or what was it used for?</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">2
types of missions required the Saturn-V launch vehicle. Let’s see them one by
one in detail;<o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">Apollo Missions:</span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; text-align: justify; vertical-align: baseline;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgjrjBTvX6CW3NXtbPjJwDUv_CfSpuDxsISHKwyMFGjmBwkRsJxQxnA1II2cc8usn5M19bR9k6FiOvDma0cTgaOsENiHwXtumhRTGS3EjSNaXSgu9GJlcfaP4lSXxw1rtRCRyQTlh_JmhQk_ZbAlwZwaZ4pRUJe4O7e4kVWcRcTLwt6wG7Fz9xDBPd6=s946" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="710" data-original-width="946" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEgjrjBTvX6CW3NXtbPjJwDUv_CfSpuDxsISHKwyMFGjmBwkRsJxQxnA1II2cc8usn5M19bR9k6FiOvDma0cTgaOsENiHwXtumhRTGS3EjSNaXSgu9GJlcfaP4lSXxw1rtRCRyQTlh_JmhQk_ZbAlwZwaZ4pRUJe4O7e4kVWcRcTLwt6wG7Fz9xDBPd6=w320-h240" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">View of the earth from the moon</td></tr></tbody></table><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">As obvious as it is, the Saturn-V was actually made
to accomplish <b>the biblical task of landing humans on the moon</b>. <o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">There were a series of <b>Apollo missions</b>, each with
different objectives that mainly revolved around studying the moon.<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">We learned in the 2<sup>nd</sup> article that, the
Saturn-V had 3 stages. However, when we are talking about the Apollo missions,
the rocket would become a <b>4-stage giant</b> as the command and service module and
the crew escape system got stacked. <o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The 4<sup>th</sup> stage contained the <b>Lunar Landing
Module</b>, <b>Service module</b>, and the <b>Command module</b>. The command module contained
the astronauts inside it.<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Let’s explore them one by one:<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><span style="font-size: medium;"><u>Service Module:</u></span><o:p style="font-size: 18pt;"></o:p></span></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjtIA1UOqbfPKBlWyLkhWFTtho6bOiXzEH_XHRGzXuECJ19P2qJS6rsv0HQLF977iGPhAusE-sMPfV2QVG9geCBHY34gyd1D_xvvcLXrsuPmI4vru0sVyHM8CNryjag7M4ujf5vR1RAIqDTiy8qIbPcZdgXwuhEB9hE_-jpDAOkR-Y6VLLXZ5-wcsAQ=s2364" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="2364" data-original-width="2340" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjtIA1UOqbfPKBlWyLkhWFTtho6bOiXzEH_XHRGzXuECJ19P2qJS6rsv0HQLF977iGPhAusE-sMPfV2QVG9geCBHY34gyd1D_xvvcLXrsuPmI4vru0sVyHM8CNryjag7M4ujf5vR1RAIqDTiy8qIbPcZdgXwuhEB9hE_-jpDAOkR-Y6VLLXZ5-wcsAQ=w317-h320" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="317" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Apollo service and command module</td></tr></tbody></table><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">The service module is the component that contained
all the <b>required cargo </b>for the use of the astronauts traveling aboard. <b>RCS
(Reaction Control System) thrusters</b> were used for the spacecraft's <b>yaw</b>, <b>pitch</b>,
and<b> roll control</b>. The<b> fuel cells</b> in the service module helped the spacecraft
with the crew's <b>electricity supply</b> and <b>water supply</b>. Apart from cargo and other
facilities, the Apollo service module sported an earth return engine <b>(AJ10-137)
</b>which was used to place the craft <b>in or out of the lunar orbit</b> or perform<b> mid-course
correction</b> and perform a <b>de-orbiting burn</b> for the command module to <b>re-enter
the earth</b>.</span></p>
<p class="MsoNormal" style="text-align: justify;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><span style="font-size: medium;"><u>Command Module:</u></span><o:p style="font-size: 18pt;"></o:p></span></b></p><p class="MsoNormal" style="text-align: justify;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEihz8tCWZrtVWX7QQZ24EtWT03sVRWFsXxwnqTg6kU0vHFK4oG7gKG6hjoSvIfNpEiClaeZsnK5CZ9aqHPMqrqkxN-eOvLlA_d_ZP_pNGZ8pC0wjmeomfAmt3XUi_nYkk-FVXw24fnMgJ2S15V-YD3suCkcLK8aw46FZPwmiQz0m1fMwWLpPoJqWheb=s1920" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="1280" data-original-width="1920" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEihz8tCWZrtVWX7QQZ24EtWT03sVRWFsXxwnqTg6kU0vHFK4oG7gKG6hjoSvIfNpEiClaeZsnK5CZ9aqHPMqrqkxN-eOvLlA_d_ZP_pNGZ8pC0wjmeomfAmt3XUi_nYkk-FVXw24fnMgJ2S15V-YD3suCkcLK8aw46FZPwmiQz0m1fMwWLpPoJqWheb=w320-h213" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The Apollo command module</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">The command module is the pod stacked over the
service module which contained the <b>crew</b>. The Apollo command carried <b>3 astronauts</b>
per single mission. The command module featured<b> RCS thrusters</b> for alignment
control, a <b>docking hatch</b> for docking with the<b> lunar lander module</b>, a <b>heat shield</b> for
atmosphere re-entry, <b>drogue chutes</b> to slow down the module while descending, the
<b>main parachute</b> for a <b>safe descent</b> and <b>splashdown</b> of the module.</span></p>
<p class="MsoNormal" style="text-align: justify;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><span style="font-size: medium;"><u>Launch Escape System:</u></span><o:p style="font-size: 18pt;"></o:p></span></b></p><p class="MsoNormal" style="text-align: justify;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiHLpCJfJ-yf-RWp2N5UtUOhfHSlaCH4-sDGUHKQIvs5Je5t3LgrxzZ5qytaI4b_joVubXng7AjWstTRypj7yfPiPfXQeJqLKXPl1p4mDqBDafiXqMPd5wjzZacUBaAWLMtwd3ofHm0BPrKcsDyoxJtJXX-qNObmzW6-28xhMouTLf5rJWlqq2fynaV=s3872" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="2592" data-original-width="3872" height="214" src="https://blogger.googleusercontent.com/img/a/AVvXsEiHLpCJfJ-yf-RWp2N5UtUOhfHSlaCH4-sDGUHKQIvs5Je5t3LgrxzZ5qytaI4b_joVubXng7AjWstTRypj7yfPiPfXQeJqLKXPl1p4mDqBDafiXqMPd5wjzZacUBaAWLMtwd3ofHm0BPrKcsDyoxJtJXX-qNObmzW6-28xhMouTLf5rJWlqq2fynaV=w320-h214" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Apollo launch escape hatch</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">The launch escape system is the tower above the
command module which was used in case of an <b>anomaly during the first phase of
the flight</b>. The basic use of a launch escape system is just to take the crew
module <b>as far as possible</b> from the rocket during a failure so that the crew onboard
doesn’t get affected by the explosion. In the modern SpaceX's </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://thedynamicfrequency.blogspot.com/2020/09/space-x-dragon-engineering.html" target="_blank">Dragon capsules</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">,
the LES is inbuild to the module, whereas, in the Apollo command module, the
LES was a separate system that relied on <b>solid motors</b> for propelling the module.
It was <b>jettisoned</b> once the spacecraft reached out of the earth’s atmosphere.</span></p>
<p class="MsoNormal" style="text-align: justify;"><b><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><span style="font-size: medium;"><u>Lunar module <i>Eagle</i>:</u></span><o:p style="font-size: 18pt;"></o:p></span></b></p><p class="MsoNormal" style="text-align: justify;"><b></b></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhduJYNVXYB-3vz_7IawZvjgQhY9NIpWVYJQp6G4iKQMwhSxMrFfPNHtwItIixsd11MnXZk9ssF622u43FV2lu0O_cGj3JIAlnZ1-gx8Z20r9LWObQzouAnU4uMd-uXK_kN2rMGZJcugubopro7mFnIJdPyzUvK3fajyEeo37IrGAwQTRLLThTZ97_h=s640" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="640" data-original-width="612" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhduJYNVXYB-3vz_7IawZvjgQhY9NIpWVYJQp6G4iKQMwhSxMrFfPNHtwItIixsd11MnXZk9ssF622u43FV2lu0O_cGj3JIAlnZ1-gx8Z20r9LWObQzouAnU4uMd-uXK_kN2rMGZJcugubopro7mFnIJdPyzUvK3fajyEeo37IrGAwQTRLLThTZ97_h=w306-h320" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="306" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Edwin Aldrin with the Eagle in Apollo-11 mission</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">The lunar module contained <b>a lander can</b> and <b>a moon
lander</b>. But we can’t see this module while the rocket launch, this is because
it is protected by <b>fairings</b> just under the service module, this is because the
module had an extremely <b>non-aerodynamic shape</b>, if left unprotected could have
caused flight instabilities inside the earth’s atmosphere and eventual mission
abortion. But again, how did the crew get inside the lunar module, well this
involved a simple <b>mid-flight maneuver</b> which is called<b> TDE (Transposition,
Docking, and Extraction)</b> maneuver, in which the command and service module
performs a <b>90</b></span><sup style="color: #0b5394; font-family: "Footlight MT Light", serif;"><b>o</b></sup><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><b> yaw</b> to align with the lunar module and propel forward
to dock and extract the module. The lunar lander included <b>landing legs</b>,
<b>thruster</b>, and <b>fuel </b>to land on the lunar surface, the lander-can contained crew
members inside it and could <b>separate from the lander to leave the lunar
surface</b>.</span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The lander-can then docked with the command module
where crew transferred from the lunar module to the command module and then the
lander-can is jettisoned.<o:p></o:p></span></p><p class="MsoNormal" style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgPip3Lka_DiEBRiOp1cakrb798YUSus9D-uLtX45Ygk_IqiFRkh5FO91ay0e4Uzqxj5BfBXZpqzNsPnI1q2ATkPda8Vo8jan9kRs4I6L_BPAk07cMTu2ELOpjsIHzirkwD_1KDSpRMzxZLIHI9vZi0vWYoZuTmMWfUMyCD4e6CkDz8Z-nw0IIljsEc=s640" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="640" data-original-width="612" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgPip3Lka_DiEBRiOp1cakrb798YUSus9D-uLtX45Ygk_IqiFRkh5FO91ay0e4Uzqxj5BfBXZpqzNsPnI1q2ATkPda8Vo8jan9kRs4I6L_BPAk07cMTu2ELOpjsIHzirkwD_1KDSpRMzxZLIHI9vZi0vWYoZuTmMWfUMyCD4e6CkDz8Z-nw0IIljsEc=w306-h320" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="306" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Lunar module docking with the Command module</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Then the spacecraft starts the journey back to
earth. The <b>AJ10 engine </b>performed the Earth transfer maneuver and the
de-orbiting burn once the craft enters the earth’s orbit. As the earth descend
phase starts, the service module is discarded and <b>burnt up</b> while re-entering.
The command module entered in a <b>butt-first alignment</b> so that the heat shield was
exposed first in order to convert <b>maximum kinetic energy</b> into <b>heat energy </b>and <b>slowed
the module down</b>. Even after that, the module was pretty fast, hence, the <b>drogue
chutes</b> were deployed to slow it down more before the <b>main chute is deployed</b>. <o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The mission was considered to be a success if <b>all
three astronauts got back and splashed down safely on the earth</b>.<o:p></o:p></span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">For ease of time management here is a table** briefly describing the objectives of each mission:<o:p></o:p></span></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">**If there is any problem viewing the table through mobile, kindly click desktop mode of your browser to view the whole table. If you are using chrome, then you will find this option by clicking the three-dot menu at the top right of your screen.</span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTable15Grid6ColorfulAccent1" style="border-collapse: collapse; border: none; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr>
<td style="border-bottom: solid #8EAADB 1.5pt; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-alt: solid #8EAADB 1.5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 5; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">Mission Type<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1.5pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-alt: solid #8EAADB 1.5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 1; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">Missions</span></b><b><span style="color: #202122; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1.5pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-alt: solid #8EAADB 1.5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 1; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">Description<o:p></o:p></span></b></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">A<o:p></o:p></span></b></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 4<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 6<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Unmanned flights of launch vehicles and the CSM, to demonstrate
the adequacy of their design and to certify safety for men." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="border-top: none; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 4; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">B<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 5<o:p></o:p></span></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Unmanned flight
of the LM, to demonstrate the adequacy of its design and to certify its
safety for men." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">C<o:p></o:p></span></b></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 7<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Manned flight to demonstrate performance and operability of the
CSM." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="border-top: none; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 4; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">C'<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 8<o:p></o:p></span></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Command and
service module manned flight demonstration in lunar orbit." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">D<o:p></o:p></span></b></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 9<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Manned flight of the complete lunar landing mission vehicle in
low Earth orbit to demonstrate the operability of all the equipment and
(insofar as could be done in Earth orbit) to perform the maneuvers involved
in the ultimate mission." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="border-top: none; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 4; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">E<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">___<o:p></o:p></span></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Manned flight of
the complete lunar landing mission vehicle in Earth orbit to a great distance
from earth." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">F<o:p></o:p></span></b></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 10<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"A complete mission except for the final descent to and landing
on the lunar surface." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="border-top: none; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 4; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">G<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 11<o:p></o:p></span></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"The initial
lunar landing mission." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">H<o:p></o:p></span></b></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 12<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 13 (planned)<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 14<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Precision manned lunar landing demonstration and systematic
lunar exploration." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="border-top: none; border: 1pt solid rgb(142, 170, 219); mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 4; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">I<o:p></o:p></span></b></p>
</td>
<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">___<o:p></o:p></span></p>
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<td style="border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Reserved for
lunar survey missions (not used)." <o:p></o:p></span></p>
</td>
</tr>
<tr>
<td style="background: rgb(217, 226, 243); border-top: none; border: 1pt solid rgb(142, 170, 219); mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 68; text-align: center;"><b><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">J<o:p></o:p></span></b></p>
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<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 15<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 16<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Apollo 17<o:p></o:p></span></p>
</td>
<td style="background: rgb(217, 226, 243); border-bottom: 1pt solid rgb(142, 170, 219); border-left: none; border-right: 1pt solid rgb(142, 170, 219); border-top: none; mso-background-themecolor: accent1; mso-background-themetint: 51; mso-border-alt: solid #8EAADB .5pt; mso-border-bottom-themecolor: accent1; mso-border-bottom-themetint: 153; mso-border-left-alt: solid #8EAADB .5pt; mso-border-left-themecolor: accent1; mso-border-left-themetint: 153; mso-border-right-themecolor: accent1; mso-border-right-themetint: 153; mso-border-themecolor: accent1; mso-border-themetint: 153; mso-border-top-alt: solid #8EAADB .5pt; mso-border-top-themecolor: accent1; mso-border-top-themetint: 153; padding: 0cm 5.4pt;" valign="top">
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: 0cm; margin-right: 0cm; margin-top: 12.0pt; margin: 12pt 0cm; mso-yfti-cnfc: 64; text-align: center;"><span style="color: #2f5496; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; mso-themecolor: accent1; mso-themeshade: 191;">"Extensive scientific investigation of Moon on the lunar surface
and from lunar orbit." <o:p></o:p></span></p>
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</tbody></table>
<p class="MsoNormal" style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiG6rivR5xuc3OvhJIMjWSaTKjICRjuuAbMpC_29cFW6b0ZjeCIYoyb7ladX6B0n6TvBGdmdy-I9XkKimP5akqS_OPQk55E842fCnjPD4rdrUFKtGHXmmvBNQQeZp7iHNSYA2DngnKFML2yKEcETCUmxFY2ckPLMymjR2XDX1zFuazbhSxDRa40U6MB=s800" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="674" data-original-width="800" height="270" src="https://blogger.googleusercontent.com/img/a/AVvXsEiG6rivR5xuc3OvhJIMjWSaTKjICRjuuAbMpC_29cFW6b0ZjeCIYoyb7ladX6B0n6TvBGdmdy-I9XkKimP5akqS_OPQk55E842fCnjPD4rdrUFKtGHXmmvBNQQeZp7iHNSYA2DngnKFML2yKEcETCUmxFY2ckPLMymjR2XDX1zFuazbhSxDRa40U6MB=w320-h270" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Eagle leaving the moon captured by the Service module</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">We will surely discuss the apollo mission in detail
in an upcoming blog, once it is uploaded the link will be available <b>here</b>.</span></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p>
<p class="MsoNormal" style="text-align: justify;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 18pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">The Skylab Mission:<o:p></o:p></span></b></p>
<p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The Skylab was the <b>first United States space
station</b>. It was launched on <b>14 May 1973</b>. Now unlike the present <b>International
Space Station</b>, the Skylab was not assembled in parts rather the whole station
was carried to orbit in a single piece by the Saturn-V rocket in its very <b>last
flight</b>. Now the Skylab was not as large as the present <b>ISS</b>, so it could easily
fit in the massive <b>fairings</b> of the Saturn-V! Wait, what! Fairings! Well yes,
not many have seen the Saturn-V with fairings, no problem here’s the picture:<o:p></o:p></span></p><p class="MsoNormal" style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhaOYaFxGwMDyN2wfrowpwNcvkFk0SwVba9sbaNnS3_bArr5QxCpUAA7bYERS1cgddnyX9hI-tFnPBVFTMLltepJJGlV7me7OkIUo9gwj0MMT_HQ3eDSGcJ_dBgB74JxYbsnWD-jeO5G3tTefEsuFjzNGYqwlYMhO4oUfUOGr6ndCYF_d55hy6IKBNX=s3000" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="3000" data-original-width="2318" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhaOYaFxGwMDyN2wfrowpwNcvkFk0SwVba9sbaNnS3_bArr5QxCpUAA7bYERS1cgddnyX9hI-tFnPBVFTMLltepJJGlV7me7OkIUo9gwj0MMT_HQ3eDSGcJ_dBgB74JxYbsnWD-jeO5G3tTefEsuFjzNGYqwlYMhO4oUfUOGr6ndCYF_d55hy6IKBNX=w247-h320" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="247" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Saturn-V in Skylab mission</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">The Skylab contained a <b>payload
shroud</b>, <b>apollo telescope</b>, <b>multiple docking adapter</b>, <b>antilock module</b>, <b>airlock
module</b>, and <b>Saturn V instrument unit</b>. This Skylab is now decommissioned though,
but when active, the <b>Apollo-Soyuz</b> program used to put humans in the Skylab.</span></p><p class="MsoNormal" style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhPjNkNV_b8YhtnUGIF-Ms-VNfOeFBVQWbIw1b_mpFsrnee8kxihJGCuGY8A04oWjry2dBOgjf_uzzgQu1lKq_HOtVOpwUV6o-NfMq75U1bRJPPyac1PJgygZ9WLj_hpxSpeaUeRvpeP05-IAPaOuIkeyZgH_LbPxfCqcwVecPdhUnhqbWzcAic9Ad6=s3000" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" border="0" data-original-height="2374" data-original-width="3000" height="253" src="https://blogger.googleusercontent.com/img/a/AVvXsEhPjNkNV_b8YhtnUGIF-Ms-VNfOeFBVQWbIw1b_mpFsrnee8kxihJGCuGY8A04oWjry2dBOgjf_uzzgQu1lKq_HOtVOpwUV6o-NfMq75U1bRJPPyac1PJgygZ9WLj_hpxSpeaUeRvpeP05-IAPaOuIkeyZgH_LbPxfCqcwVecPdhUnhqbWzcAic9Ad6=w320-h253" title="Saturn-V for Dummies Part-4: Missions The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">An Apollo module docked with the Skylab</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">After spending <b>2249 days</b> in orbit and
making <b>34981 orbits</b> around the planet the Skylab was deorbited on <b>11 July 1979</b>.
This happened due to the delay in the <b>space shuttle program</b>. Now get the
physics here, not everything is <b>ideal</b> in space, the orbit in which a particular
object is placed eventually <b>decays</b>, so the Skylab needed to be<b>
re-boosted</b> to continue its orbit (yes, the current ISS also does that). So, by
this way Skylab came to an end, it deserves a <b>dedicated article</b> for itself so
we’ll surely cover it in the future. But in our next article, we'll conclude this series by answering some fundamental questions like <b><a href="https://thedynamicfrequency.blogspot.com/2022/02/saturn-v-for-dummies-part-5-conclusion.html" target="_blank">"Why have humans not returned to the moon till now?"</a> </b>So, stay tuned.</span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">If you like our articles then please <b>follow this blog</b> to get the <b>latest updates directly to you</b> <b>(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"><o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><b style="background: 0px 0px; border: 0px; color: #474747; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">To know the basics of the quantum world, astronomy and space exploration you can </b><b style="background: 0px 0px; border: 0px; color: #474747; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">check out the book <a href="https://www.amazon.in/dp/B07TBSP6JT" style="background: 0px 0px; border: 0px; color: #e94bc3; outline: 0px; padding: 0px; text-decoration-line: none; transition: color 0.17s ease 0s; vertical-align: baseline;" target="_blank">"Through the wormhole" on amazon kindle</a> by Ratnadeep Das Choudhury.</b></p><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"></span></span><br /><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">Thanks for Reading!!!</span></div><p class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px; outline: 0px; padding: 0px; vertical-align: baseline;"></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span></p><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;">Check out our other blog posts!!!</span></div></div><div style="background: 0px 0px; border: 0px; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="background: 0px 0px; border: 0px; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></div></div><div style="background: 0px 0px; border: 0px; margin: 0cm 0cm 0.0001pt; outline: 0px; padding: 0px; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: times, "times new roman", serif; outline: 0px; padding: 0px; vertical-align: baseline;"><o:p></o:p></span></div></div><div style="background: 0px 0px; border: 0px; 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border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></i></div></div><div style="background: 0px 0px; border: 0px; font-family: "Times New Roman"; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;">See you Again!!!!!</span></div><div><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></span></div></div></div><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; vertical-align: baseline;"><span style="color: red; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p align="right" class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: right; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"> </span><span style="background: 0px 0px transparent; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background-color: transparent; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt;">-Ayushman Dash</span></p><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; font-family: "times new roman"; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"> Chief Space Exploration Writer</span></div><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">The Dynamic Frequency</span></div></div>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><o:p></o:p></span></p>
<p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; mso-margin-top-alt: auto; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-90846171002842958532022-01-22T22:41:00.012+05:302022-02-23T00:06:14.206+05:30Saturn-V for Dummies Part-3: The Engines<p class="MsoNormal" style="text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjyPwJ-9IClmzqUDvMrgEq4D208GFZ4CVBZFfIbx-iLmPAIP0N6IeIX-5HgfgaFQhMjnJGOEvWDazkAOH4iu7loKqmOvuU0zQBemEFseLw3sdXiEa9UpIYRPalG-QdX8QnuqPMsWfpu5_-3LM6IjLYPJIGdzmg3xArbrwa_SWN3tNlYjoxmVqxpaExL=s6000" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" border="0" data-original-height="4000" data-original-width="6000" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEjyPwJ-9IClmzqUDvMrgEq4D208GFZ4CVBZFfIbx-iLmPAIP0N6IeIX-5HgfgaFQhMjnJGOEvWDazkAOH4iu7loKqmOvuU0zQBemEFseLw3sdXiEa9UpIYRPalG-QdX8QnuqPMsWfpu5_-3LM6IjLYPJIGdzmg3xArbrwa_SWN3tNlYjoxmVqxpaExL=w320-h213" title="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Inside the F-1 engine</td></tr></tbody></table><p></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: large;"><b>H</b></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">ey again everybody, in our journey about studying
the mighty Saturn-V, till now we saw the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt1-history.html" target="_blank">history of its development</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> and
got a basic </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturnV-for-dummies-pt2-machine-overview.html" target="_blank">overview of the rocket</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"> if you haven’t read them yet, I
highly recommend going through them now as the previous blogs play a vital role
in setting up the context of this blog. But what is this blog really about? As
is evident from the topic mentioned above, we’ll be learning about the engines
today. <b>A vehicle can only be called a rocket if it has a rocket engine</b>, rocket
engines are <b>reaction engines</b> meaning that spew out fluid that propels the
rocket as a result of the reaction of that spewing. Now the engines need to
spew out fluid with a certain velocity to produce force/thrust. The force shall
be able to lift the rocket off the ground. From all that we know about the
Saturn-V till now, just try to imagine the enormity of the engines that it
would have required. Now let’s dive in without much further ado.</span></p><p class="MsoNormal" style="text-align: justify;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light",serif; font-size: 24pt; line-height: 107%; mso-ansi-language: EN-US;"> </span></b></p><p class="MsoNormal" style="text-align: justify;"><b><span lang="EN-US" style="color: red; font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="font-size: x-large;">The Engines: Or how the Rocket flew to the Moon ?</span><span style="font-size: 24pt;"><o:p></o:p></span></span></b></p><p class="MsoNormal" style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">There were 2 types of engines used throughout the
rocket stages. Let’s see them one by one in detail;</span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light", serif; padding: 0cm;"><span style="font-size: large;">F-1
engine:</span></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgBkYtD5XYTeD35BrRmrU7SglnY72loRg8elUnnl1DhlImukZdZCHIFSR9MjhYljBqDrAnBVSevF7fTbQl9oM-ZhC5VqpoGJ9bNmJ4LpzRymt4Kvzky7JOOZ7CqaI8q_sit30WdimQwk0mvpllQaMxw_nZT4s1WNKsyasDTAwmnaQOeMSHA4M-9OZEe=s1200" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" border="0" data-original-height="900" data-original-width="1200" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEgBkYtD5XYTeD35BrRmrU7SglnY72loRg8elUnnl1DhlImukZdZCHIFSR9MjhYljBqDrAnBVSevF7fTbQl9oM-ZhC5VqpoGJ9bNmJ4LpzRymt4Kvzky7JOOZ7CqaI8q_sit30WdimQwk0mvpllQaMxw_nZT4s1WNKsyasDTAwmnaQOeMSHA4M-9OZEe=w320-h240" title="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The F-1 engine on display at the INFINITY science center</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
F-1 engine was developed by Rocketdyne as one of the 2 engines (<b>F-1 and E-1</b>)
to meet a <b>1955</b> <b>U.S. Airforce requirement</b> for a large rocket
engine. The E-1 engine however was scrapped as it was seen as a technological
dead end. The development of the F-1 engine continued further as <b>NASA</b>
appreciated its usefulness. <o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
engine used <b>RP-1</b> as the fuel and <b>Liquid Oxygen (Lox) </b>as the oxidizer. Both
oxidizer and fuel were mixed in a <b>2.27:1</b> ratio and were burnt in the
combustion chamber. <o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">Many
rocket engines use <b>turbo-pumps</b> to suck the fuel and oxidizer from the
<b>pressurized tanks</b> and inject them into the <b>combustion chamber</b>, to spin the pump,
the F-1 engine followed a <b>gas generator path </b>in which a small amount of
fuel and oxidizer is burnt in a <b>pre-burner chamber</b> which spins the turbine, the
exhaust is usually dumped on board. Then the fuel and oxidizer mixture is burnt
in the combustion chamber and expelled through the bell nozzle. This mechanism
is similar to the one we see in <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/spaceX-merlin-intro-to-turbopump.html" target="_blank">SpaceX’s Merlin Engine</a></b> but the
difference is, the merlin dumps the exhaust onboard whereas the F-1 uses the
<b>“cooler”</b> pre-burner exhaust as a <b>heat exchanger </b>to stop melting of the nozzle
by over-heating. The <b>“hotter”</b> pre-burner exhaust is then dumped with the <b>main
exhaust</b>! Name a more ingenious cooling technique used in any other rocket
engine, I’ll wait.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEicac8iwuR6q02IJFYQETNsydkNIp28wXdGSsjrtUrqb3lYmicPTwPa-eTKfy5176QAWl4oauy14rL_ZKcZuqwosSZNduQrLzeIoA8FS-8HvYvh0hFQhG-Ntb53XhzScfWKDDNs7Q9CM-MldC9NbrPhx9Wmm6ybqOM6yqR9obH5WG6VYvM5xxUOyrUp=s898" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" border="0" data-original-height="898" data-original-width="732" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEicac8iwuR6q02IJFYQETNsydkNIp28wXdGSsjrtUrqb3lYmicPTwPa-eTKfy5176QAWl4oauy14rL_ZKcZuqwosSZNduQrLzeIoA8FS-8HvYvh0hFQhG-Ntb53XhzScfWKDDNs7Q9CM-MldC9NbrPhx9Wmm6ybqOM6yqR9obH5WG6VYvM5xxUOyrUp=w261-h320" title="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" width="261" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Gas-Generator cycle Engine (F-1 & J-2)</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
F-1 produced <b>6770kN</b> of thrust at <b>sea level</b> and <b>7770kN </b>of thrust in <b>vacuum</b>.
It had a <b>thrust-to-weight</b> ratio of <b>94:1</b> (the thrust-to-weight ratio is a
performance measure of a rocket engine obtained by dividing thrust by the
weight of the vehicle), <b>specific impulse</b> (specific impulse is the ability of an
engine to produce an acceleration of <b>9.81N</b> of force<sup> </sup>for a certain duration
of time for <b>1kg of fuel </b>and is usually measured in seconds, it is like the <b>mileage </b>of the rocket
engine) of the engine was <b>263s at seal level </b>and <b>304s in vacuum</b>. <o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">This
makes the F-1 engine the <b>most powerful gas-generator cycle engine</b> ever made. In
the Saturn-V vehicle, <b>5 </b>F-1 were used in the booster <b>(S-IC)</b> stage and operated
for a total of <b>150s-163s</b> which propelled the vehicle to the first <b>61km </b>of its ascent.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light", serif; padding: 0cm;"><span style="font-size: large;">J-2
engine:</span><span style="font-size: 13.5pt;"><o:p></o:p></span></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgHFxFfhzjOVVaE64j_-ERoGdSCGqcoOHUv1gCTIjjNCuucTyYFYPfoDMlL0lb6UsHuXGGCfkkog7CCepXb0hl_-wXG9RZoe2XHAUdM4y8PdZ-PNEpd8H8Fimxm6qAnoh7FkSVR4fzGv_91URzDzb3KEiBi0zsUm5I-CXeLTSEks14wdPyzMmGQIEdo=s899" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" border="0" data-original-height="899" data-original-width="574" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEgHFxFfhzjOVVaE64j_-ERoGdSCGqcoOHUv1gCTIjjNCuucTyYFYPfoDMlL0lb6UsHuXGGCfkkog7CCepXb0hl_-wXG9RZoe2XHAUdM4y8PdZ-PNEpd8H8Fimxm6qAnoh7FkSVR4fzGv_91URzDzb3KEiBi0zsUm5I-CXeLTSEks14wdPyzMmGQIEdo=w204-h320" title="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" width="204" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The J-2 engine on display at London Science Museum</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
J-2 engine was a <b>vacuum engine</b> meaning that its performance was better at
higher altitudes (outside the earth’s atmosphere) than when operated inside the
atmosphere. As a result of which the primary objective of the engine was upper
atmosphere manoeuvres and <b>Earth Orbit Insertion</b> or <b>Trans Lunar Injection</b>, hence
the engine was designed to be <b>restartable</b>. <o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
engine burned <b>liquid hydrogen (LH<sub>2</sub>)</b> as fuel and <b>liquid oxygen (LO<sub>2</sub>)</b>
as the oxidizer with a <b>5.5:1 oxidizer to fuel ratio</b> in the combustion chamber. The
engine’s turbopump cycle is the same as that of the F-1 engine i.e., <b>closed gas
generator cycle</b> where the cooler pre-burner exhaust was used as a heat
exchanger for the nozzle and then was dumped with the main exhaust.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
engine produced a thrust of <b>1033.1kN</b> of thrust in <b>vacuum</b> and <b>486.2kN at sea
level</b>. In a vacuum, the engine had a specific impulse of <b>421s </b>and at sea level,
it was <b>200s </b>which makes sense as it was never made to be used as a sea-level
engine. The thrust-to-weight ratio of the engine was <b>73:18</b>.</span><b><span style="border: 1pt none windowtext; color: lime; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;"><o:p></o:p></span></b></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">In
the Saturn-V, <b>5 </b>of J-2 engines were used in the <b>S-II second stage</b> and <b>1 </b>J-2 was
used in the<b> S-IVB 3<sup>rd</sup> stage</b>. In the 2<sup>nd</sup> stage, the engine
was burnt for about <b>360s</b> propelled the vehicle to an altitude of <b>185km</b>, and
provided a near orbital velocity of about <b>25000km/h</b>. In the 3<sup>rd</sup> stage,
the J-2 burnt for about <b>165s </b>and made the <b>EOI</b> and <b>TLI</b> maneuvers.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjfd_cqRZEH-xLdPhTe0jXBVXvDEWQ2_pOy1IIT9g9O8A28aZO86opvWzvgJJfGF9UiKQ2TbQ0oSfcIus_rpYiJoVkwQGE_vdRwqryl1zx21IjNfJaYyEPb1nmyPq9daCKoLStj456C2sWL80LK4a9nqFZXSTyyAVqm54jVGntIHan_86OcKOD09YxD=s887" style="margin-left: auto; margin-right: auto;"><img alt="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" border="0" data-original-height="887" data-original-width="665" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjfd_cqRZEH-xLdPhTe0jXBVXvDEWQ2_pOy1IIT9g9O8A28aZO86opvWzvgJJfGF9UiKQ2TbQ0oSfcIus_rpYiJoVkwQGE_vdRwqryl1zx21IjNfJaYyEPb1nmyPq9daCKoLStj456C2sWL80LK4a9nqFZXSTyyAVqm54jVGntIHan_86OcKOD09YxD=w240-h320" title="Saturn-V for Dummies Part-3: Engines The Dynamic Frequency" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The J-2X engine</td></tr></tbody></table><p></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">The
J-2 engine was one of the most reliable engines NASA ever had. The next-gen J-2
(<b>J-2X</b>) engine was considered for the <b>Earth Departure Stage</b> of the <b>Space Shuttle
program</b>.</span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;">As
we saw the stats of the engine used in the Saturn-V vehicle are incredible and
straight out insane. Well, then you might be wondering what was the purpose of
the vehicle? Why did we need such incredible power? We already know it’s the
<b>Apollo program</b>, yes, but was the vehicle only used for <b>taking humans to the moon</b>?
Well, <b>no</b>. So, this sets up the context for our next article. In the next
article, we’ll be learning about the <b><a href="https://thedynamicfrequency.blogspot.com/2022/02/saturn-v-for-dummies-pt4-missions.html" target="_blank">missions and accomplishments</a></b> of this beast
of a launch vehicle. Once it is published, the link will be available here.<o:p></o:p></span></p><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-border-alt: none windowtext 0cm; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN; padding: 0cm;">If you like our articles then
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mso-fareast-language: EN-IN; padding: 0cm;"> </span><span style="background: 0px 0px transparent; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background-color: transparent; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt;">-Ayushman Dash</span></p><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; font-family: "times new roman"; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"> Chief Space Exploration Writer</span></div><div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">The Dynamic Frequency</span></div></div><p class="MsoNormal" style="background: white; line-height: normal; margin-bottom: 0cm; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-IN;"> </span></p><p class="MsoNormal" style="text-align: justify;">
</p><p class="MsoNormal" style="text-align: justify;"><span lang="EN-US" style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-ansi-language: EN-US;"> </span></p>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com1tag:blogger.com,1999:blog-3578987301814134971.post-77854991749735569162022-01-14T21:35:00.019+05:302022-02-23T00:06:37.291+05:30SATURN-V for Dummies Part-2: The Machine Overview<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: "Footlight MT Light", serif;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg-TEHoN8f5ltphmuEFGhXMqRYf5vafmq9490rwOajLAmyuzG6uHFNRTiCrdmAafj3PnXXYGFLVhq4-GbYgwCpq6tMxIa6AuKqhkB9b6JzzBmcZf4nwA94t72qybbzf_wy3xTlgOW-E-kCeVc3ivYbf8LvqjOPi7uVpLd6NIuFTVgZleacRW7cqZOWk=s2048" style="margin-left: 1em; margin-right: 1em;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="1536" data-original-width="2048" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEg-TEHoN8f5ltphmuEFGhXMqRYf5vafmq9490rwOajLAmyuzG6uHFNRTiCrdmAafj3PnXXYGFLVhq4-GbYgwCpq6tMxIa6AuKqhkB9b6JzzBmcZf4nwA94t72qybbzf_wy3xTlgOW-E-kCeVc3ivYbf8LvqjOPi7uVpLd6NIuFTVgZleacRW7cqZOWk=w320-h240" title="SATURN-V for Dummies Part-2: The Machine Overview" width="320" /><br /></a></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: #0b5394;"><span><b><span style="font-size: x-large;"><br /></span></b></span></span></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: #0b5394;"><span><b><span style="font-size: x-large;">H</span></b><span style="font-size: large;">appy Makar Sankranti/Pongal and t</span></span><span style="font-size: large;"><span>hank you, readers, for the tremendous
love and support in my previous article on the History of Saturn-V. If you
haven’t read that yet, I would highly recommend giving it a read, this will help you to
get the context of this </span><span><span>article (</span><b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt1-history.html" target="_blank">click here to read it</a></b><span>)</span></span><span>. In this article, I
will be giving you the machine overview and try to answer the question, what
this Saturn-V actually is. So, without much further ado, let’s get to the
point.</span></span></span></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: #0b5394;"><span style="font-size: large;"><span><br /></span></span></span></span></div><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="font-family: "Footlight MT Light", serif;"></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: #0b5394;"><span style="font-size: medium;"><br /></span></span></span></p><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span><span lang="EN-US" style="color: red; font-family: "Footlight MT Light", serif; font-size: x-large;"><b>The Machine Overview: Getting the Rough Idea</b></span></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span><span lang="EN-US" style="color: red; font-family: "Footlight MT Light", serif; font-size: x-large;"><b><br /></b></span></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif; font-size: large;">Saturn-V is a <b>3-stage</b> vehicle, with each stage namely; <b>S-IC first
stage</b>, <b>S-II second stage</b>, <b>S-IVB third stage</b>, and <b>Instrument unit</b>. If you find
the stage names confusing, don’t worry I’ll also explain the logic behind the nomenclature. But why did the rocket need 3 stages to get to the moon? The simple answer is <b>mass management </b>and <b>gravity</b> or simply the <b>Tyranny of the Rocket Equation</b>; a massive amount of fuel is needed to get to high altitudes, <b>85-90%</b> of the rocket is just the <b>fuel</b>, and the rest <b>10-15%</b> comprises the <b>dry mass</b>. In that case, it is more efficient to just expend a stage after the fuel is used up. though there are many on-paper concepts of <b>Single Stage To Orbit (SSTO)</b> rockets, none of them have flown yet (SSTOs will be covered in a future article). So, staging seems a <b>wiser and better idea </b>to get to orbit.</span></span></div><p></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="color: #0b5394; font-family: Footlight MT Light, serif; font-size: medium;"><br /></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgrX3V--Emtyt91XSbSuHBFms_X5a4HObarHCsckAeFeaUTUwcUbBk5opHeCQQw-VKpgMsFN3XAk1TSCAw7MfhkBJUwHmXrMjvzOVb68StNgLPu-y-H8ROtlO-cRs7Va-iPiZhtz-JYdpUtOqn-WC-HndHOwsog53gRap9UQfcLcLS7TdOQgpXTLjM_=s2717" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="1803" data-original-width="2717" height="212" src="https://blogger.googleusercontent.com/img/a/AVvXsEgrX3V--Emtyt91XSbSuHBFms_X5a4HObarHCsckAeFeaUTUwcUbBk5opHeCQQw-VKpgMsFN3XAk1TSCAw7MfhkBJUwHmXrMjvzOVb68StNgLPu-y-H8ROtlO-cRs7Va-iPiZhtz-JYdpUtOqn-WC-HndHOwsog53gRap9UQfcLcLS7TdOQgpXTLjM_=w320-h212" title="SATURN-V for Dummies Part-2: The Machine Overview" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Saturn-V with all its stages in Houston, USA</td></tr></tbody></table><br /><p></p><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><b style="color: #04ff00; font-family: "Footlight MT Light", serif;"><span style="font-size: large;">S-IC First Stage:</span></b></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">The 1</span><sup style="color: #0b5394; font-family: "Footlight MT Light", serif;">st</sup><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> stage was <b>42m</b> tall, <b>10m</b> in
diameter, and provided <b>33000kN (33MN)</b> of thrust to get the rocket through the
first <b>61km</b> of ascent. The dry mass of the stage was <b>131000kg</b> and, once fully fueled it weighed </span></span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">2.3 million kg</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">. </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Rocket-grade kerosene (RP-1)</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> </span><span><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">was used as the fuel for this
particular stage and </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">liquid oxygen (Lox)</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> was used as the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">oxidizer</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">. The stage
had </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">five F-1 engines </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">in a </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">quincunx </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">arrangement. The center engine was fixed
whereas the four outer engines could be </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">gimballed</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> using </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">hydraulic systems</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> to
control the rocket.</span></span></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><p style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;"><b>The Boeing Company</b> was awarded the contract to
manufacture the S-IC first stage on <b>December 15, 1961</b>. By then, the general
design of the first stage had been decided by the engineers at the <b>MSFC</b>. The MSFC engineers also built the <b>first 3 test stages</b> (S-IC-T, S-IC-S, S-IC-F), and the <b>first
two flight models</b> (S-IC-1 and -2). The main place of manufacture was the
<b>Michoud Assembly Facility, New Orleans</b>. Wind tunnel testing took place in
<b>Seattle</b> and the machining of the tools needed to build the stages at <b>Wichita,
Kansas</b>.</span></span></span></p><span style="font-size: large;"><div style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">It took around </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">9 months </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">to build the tanks and </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">14
months </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">to complete a stage. The timeline may feel absurd because today, </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">SpaceX
</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">can manufacture its </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;"><a href="https://thedynamicfrequency.blogspot.com/2021/11/spaceX-merlin-introduction.html" target="_blank">Falcon-9</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> rocket in under </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">3 months</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">, but we need to keep
in mind the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">scale</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> and the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">time</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> of manufacture. The rocket was way bigger and
way more capable than an F9 and again it was being manufactured in the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">1960s</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">
and </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">70s</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">, a time when </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">automated manufacturing</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> basically did not exist.</span></div></span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><p style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif;">The largest and heaviest single component of the S-IC was the <b>thrust
structure</b>, with a mass of <b>22000kg</b>.</span><span style="font-family: "Footlight MT Light", serif;"> It
was designed to support the thrust of the five engines and <b>redistribute</b> it
evenly across the <b>base</b> of the rocket. There were <b>four anchors</b> which held down
the rocket as it built thrust. These were among the <b>largest aluminum forgings</b>
produced in the <b>U.S.</b> at the time, <b>4.3m long</b> and <b>816kg </b>in weight.
The four stabilizing fins withstood a temperature of <b>1100 °C</b>.<br /><o:p></o:p></span></span></span></p><span style="font-size: large;"><div style="text-align: justify;"><span style="background-color: white; color: #0b5394; font-family: "Footlight MT Light", serif;">Now, S-I means </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Stage-1</b><span style="background-color: white; color: #0b5394; font-family: "Footlight MT Light", serif;">, and C means, it is the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">third design iteration</b><span style="background-color: white; color: #0b5394; font-family: "Footlight MT Light", serif;"> of the
stage and hence the name S-1C.</span></div></span></div><p></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="color: #0b5394; font-family: Footlight MT Light, serif; font-size: medium;"><br /></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj5bQRJUoYSR0VkEG1JQfKH1HIBudQtQMR1Tb6HkGACImdt-mdmCIVUuOIW81Y622hkLgtrEyUY609TTxRkFvEO9Wsr5sC036jJIH55nuurkiPJxWHdy1nGwO-MR8rndF7qp2peZt_e05oxhc9cEkvX8KiPSuORvtRJN2YqwPjEZ5bznHXv8ciHeZOP=s4608" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="3456" data-original-width="4608" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEj5bQRJUoYSR0VkEG1JQfKH1HIBudQtQMR1Tb6HkGACImdt-mdmCIVUuOIW81Y622hkLgtrEyUY609TTxRkFvEO9Wsr5sC036jJIH55nuurkiPJxWHdy1nGwO-MR8rndF7qp2peZt_e05oxhc9cEkvX8KiPSuORvtRJN2YqwPjEZ5bznHXv8ciHeZOP=w320-h240" title="SATURN-V for Dummies Part-2: The Machine Overview" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">S-IC stage on display at the Kennedy Space Centre</td></tr></tbody></table><br /><p></p><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span><span lang="EN-US" style="color: #04ff00; font-family: "Footlight MT Light", serif; font-size: large;"><b>S-II Second Stage:</b></span></span></div><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;">The 2<sup>nd</sup> stage was about <b>25m</b> tall,
<b>10m</b> in diameter, and provided <b>4,400kN</b> of thrust to accelerate the Saturn-V
through the <b>upper atmosphere</b>. The dry mass of the stage was <b>36200kg</b> and, when fully fueled, it weighed </span></span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">480000kg</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">. </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Liquid hydrogen (LH<sub>2</sub>)</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> </span><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">was used as the fuel in
this stage and liquid oxygen (Lox) was used as the oxidizer. The stage had </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">five
J-2 engines</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> in a quincunx pattern. Similar to the S-IC; the center engine was fixed and the outer four were hydraulically gimballed. </span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><br /></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">North American Aviation</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> was awarded the contract to
manufacture S-II second stage on </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">11<sup>th</sup> September 1961</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> (Note: it
is the same company that was awarded the contract for the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Apollo Command /
Service Module</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">). They were also provided with the manufacturing plant at </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Seal
Beach, California</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> by the government.</span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><br /></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">Interestingly, this stage when fully loaded; had only </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">7.6%</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> hardware, and the rest </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">92.4%</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> comprised LH</span><sub style="color: #0b5394; font-family: "Footlight MT Light", serif;">2</sub><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> and Lox. The S-II stage had some very brilliant ways to contain highly volatile liquid hydrogen and even prevent turbulence due to fuel sloshing. We will surely cover them in detail in a future article.</span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><br /></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">S-II simply means </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Stage-2</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">, the meaning is quite
literal.</span></span></p><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif; font-size: large;"><br /></span></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjYf2U4g-kAvz3JDbNHKuzHCKpNYSPz_RpGgk9EzIMfnF3syosf5g-s307A1x690NyUrcFFP7Zxpl2WY7tymFtFKXtXzr52zMPwcjZkTNyFXe6W-Lbrz4hFDjC7yJrhcSx2i0I58Ci69ezooluIF5sgDQCJjbZPpOrg4zneFSiCkvizN6pOVlGqB2BZ=s4032" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="3024" data-original-width="4032" height="240" src="https://blogger.googleusercontent.com/img/a/AVvXsEjYf2U4g-kAvz3JDbNHKuzHCKpNYSPz_RpGgk9EzIMfnF3syosf5g-s307A1x690NyUrcFFP7Zxpl2WY7tymFtFKXtXzr52zMPwcjZkTNyFXe6W-Lbrz4hFDjC7yJrhcSx2i0I58Ci69ezooluIF5sgDQCJjbZPpOrg4zneFSiCkvizN6pOVlGqB2BZ=w320-h240" title="SATURN-V for Dummies Part-2: The Machine Overview" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">S-II stage on display at the Kennedy Space Centre</td></tr></tbody></table><span style="font-size: large;"><br /></span><p></p><div style="text-align: left;"><span style="font-size: large;"><span><span lang="EN-US" style="color: #04ff00; font-family: "Footlight MT Light", serif;"><b>S-IVB Third Stage:</b></span></span></span></div><div style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">The 3</span><sup style="color: #0b5394; font-family: "Footlight MT Light", serif;">rd</sup><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> stage was </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">17.8m</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> tall, </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">6.6m </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">in diameter and could provide a maximum thrust of </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">890kN</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> (S-IVB </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">200 series</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">) or </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">1MN</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> (S-IVB </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">500 series</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">). The stage weighed </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">13500kg</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> when empty and </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">123000kg</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> when fully fueled. </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Similar to the S-II stage</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"> it used LH</span><sub style="color: #0b5394; font-family: "Footlight MT Light", serif;">2 </sub><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">and Lox fuel and oxidizer respectively. The stage used only </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">one J-2 engine </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">and was restartable. For lunar missions, it was </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">fired twice</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">: first for </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">Earth orbit insertion </b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">after </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">second stage cutoff (SECO)</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">, and then for </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif;">translunar injection (TLI)</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">.</span></span></div><p style="text-align: justify;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif; font-size: large;">This stage evolved from the<b> upper stage of the
Saturn 1 rocket</b> and was the <b>first stage</b> of the Saturn-V to be designed.
NASA awarded the manufacturing contract to <b>Douglas Aircraft Company</b> on <b>19
April 1960</b>. At the same time, it was decided to make the <b>Saturn-1b</b> that
would also use the S-IVB as the <b>second stage</b> and could be used for <b>testing</b> the
<b>Apollo spacecraft</b> in the <b>Earth orbit</b>.</span></span></p><div style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;">S-IVB was originally meant to be the <b>fourth stage</b>
of the <b>C-4 rocket</b> (<b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt1-history.html" target="_blank">covered in the previous article</a></b>), and the ‘B’ because, it
was the <b>second manufacturing iteration</b>, the previous version used a cluster of
<b>six J-2 engines</b>. Hence the name, S-IVB.</span></span></span></div><div style="text-align: justify;"><span style="color: #0b5394; font-size: medium;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;"><br /></span></span></div><p></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEivMXk1e5sl7DhUtLRa0vHes-jqRF9tAbjNrF28wPoQjsp8919lS9ZRqE0Honp8Bc0IWgbHsitYT7TDnuiohu8RycB87hCesPz7j5VLZwO1AG0wBQ9FZq-jKuGJpUVyiFs3f_MFCwKXlzGQpTYv7y_oa9qTnLO075SsBHnSbCC2GvvaaplIQsZM8Grn=s4170" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="4170" data-original-width="4132" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEivMXk1e5sl7DhUtLRa0vHes-jqRF9tAbjNrF28wPoQjsp8919lS9ZRqE0Honp8Bc0IWgbHsitYT7TDnuiohu8RycB87hCesPz7j5VLZwO1AG0wBQ9FZq-jKuGJpUVyiFs3f_MFCwKXlzGQpTYv7y_oa9qTnLO075SsBHnSbCC2GvvaaplIQsZM8Grn=w317-h320" title="SATURN-V for Dummies Part-2: The Machine Overview" width="317" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">S-IVB stage separation in Apollo-7 mission</td></tr></tbody></table><br /><p></p><div style="text-align: left;"><span><span lang="EN-US" style="color: #04ff00; font-family: "Footlight MT Light", serif; font-size: large;"><b>Instrument Unit:</b></span></span></div><p></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="color: #04ff00; font-family: Footlight MT Light, serif; font-size: large;"><b><br /></b></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhqjkArCM61VonOvljxFu-IKtuleJZe0DEZcClCqHVwiF-7FboKHZD268Tl85ouoK-_7FZWfAxgz-csfDi28OZftP2vzJn-DIiHivQRk4sQC0765kO__Zkwb3jcqG3onXTnrAdzd-3MlKIb5NKSrFrDXJ_0v5DQSbolhH3RFx0ZvBoaOW_ioFckn4LP=s3000" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-2: The Machine Overview" border="0" data-original-height="2292" data-original-width="3000" height="244" src="https://blogger.googleusercontent.com/img/a/AVvXsEhqjkArCM61VonOvljxFu-IKtuleJZe0DEZcClCqHVwiF-7FboKHZD268Tl85ouoK-_7FZWfAxgz-csfDi28OZftP2vzJn-DIiHivQRk4sQC0765kO__Zkwb3jcqG3onXTnrAdzd-3MlKIb5NKSrFrDXJ_0v5DQSbolhH3RFx0ZvBoaOW_ioFckn4LP=w320-h244" title="SATURN-V for Dummies Part-2: The Machine Overview" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Manufacturing the Instrument Unit</td></tr></tbody></table><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;"><br /></span></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;">The instrument unit was built by <b>IBM</b> and was placed on top of the third stage. This was the <b>brain</b> of the whole rocket and <b>controlled all the operations</b> of the rocket just before liftoff until the S-IVB stage was <b>discarded</b>. Now guess what, we have a whole article fully dedicated to the sheer brilliancy of this 60s old school computer. <b><a href="https://thedynamicfrequency.blogspot.com/2020/04/america-moon-Apollo-computer.html" target="_blank">Go, check that out</a></b>!!</span></span><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;"><br /></span></span></span></p><p style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394;"><span lang="EN-US" style="font-family: "Footlight MT Light", serif;">These were the 3 stages of the mighty Saturn-V rocket, once all the stages were built, they were <b>vertically integrated</b> at the <b>Vehicle Assembly Building (VAB)</b> at<b> NASA’s Kennedy Space Centre, Florida</b>. It took an average of <b>2 years to build a single Saturn-V</b> and needed a <b>giant crawler-transporter </b>for carrying the beast to the <b>launchpad</b>.</span></span></span></p><p style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">The project did cost around <b>6.417 billion USD</b>, with <b>cost per launch</b> leading up to a whopping <b>185 million USD</b> (well today it would have cost <b>$1.33 billion</b>). If we take the <b>inflation rate</b> as <b>7.19</b>, then this whole project would cost around <b>$46.11 billion (as of 2021)</b> to put things into perspective NASA’s <b>2021 budget</b> is <b>$23.3billion,</b> and in the 60s and 70s Apollo wasn’t the only program that NASA was funding, that's a lot of money!</span><span style="font-family: "Footlight MT Light", serif;"><o:p></o:p></span></span></p><p style="text-align: justify;"><span style="font-size: large;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;">Though these accomplishments seem nuts, trust me these are nothing compared to the glorious <b>fire-breathing bells</b> that the rocket had underneath it. Stay tuned because in my next article we'll be <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturn-v-for-dummies-pt-3-the-engines.html" target="_blank">stripping down the engines of Saturn-V</a></b>. Get ready to be blown away by the marvels of modern engineering.</span><span style="font-family: "Footlight MT Light", serif;"><o:p></o:p></span></span></p><p></p><p style="background: 0px 0px white; margin: 0cm; outline: 0px; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light", serif; padding: 0cm;"><span style="font-size: large;">If you like our articles then please <b style="background-position: 0px 0px; outline: 0px;">follow this blog</b> to get the <b style="background-position: 0px 0px; outline: 0px;">latest updates directly to you</b> <b style="background-position: 0px 0px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span></span></p><p style="background: 0px 0px white; margin: 0cm; outline: 0px; vertical-align: baseline;"><span style="border: 1pt none windowtext; color: #0b5394; font-family: "Footlight MT Light", serif; padding: 0cm;"><span style="font-size: large;"><b style="background-position: 0px 0px; outline: 0px;"><br /></b></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"><b style="background: 0px 0px; border: 0px; color: #474747; outline: 0px; padding: 0px; vertical-align: baseline;">To know the basics of the quantum world, astronomy, and space exploration you can check out the book <a href="https://www.amazon.in/dp/B07TBSP6JT" style="background: 0px 0px; border: 0px; color: #e94bc3; outline: 0px; padding: 0px; text-decoration-line: none; transition: color 0.17s ease 0s; vertical-align: baseline;" target="_blank">"Through the wormhole" on amazon kindle</a> by Ratnadeep Das Choudhury.</b></span></span></p><p style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="color: #474747; font-family: footlight mt light, serif;"><span style="font-size: 18px;"><b><br /></b></span></span></p><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;">Thanks for Reading!!!</span></div><p class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "Footlight MT Light", serif; font-size: 13.5pt; 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outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span>-Ayushman Dash</span></div></div><div style="background: 0px 0px; border: 0px; color: black; font-family: "times new roman"; font-size: medium; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"> Chief Space Exploration Writer</span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background-color: initial; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">The Dynamic Frequency</span></div></div></span></div></div></div><p style="text-align: left;"></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-size: medium;"></span></p><p style="text-align: left;"></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-size: medium;"><span style="background: white; font-family: "Footlight MT Light", serif;"></span></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-size: medium;"></span></p><p style="text-align: left;"></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-size: medium;"></span></p><p style="text-align: left;"></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-size: medium;"></span></p><p style="text-align: left;"></p><div class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"></div><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; border: 0px; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"></span></span></div><p class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px; outline: 0px; padding: 0px; vertical-align: baseline;"></span></p><p class="MsoNormal" style="line-height: normal; 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</p><p class="MsoNormal" style="text-align: left;"><span style="font-family: "Footlight MT Light",serif;"><span style="color: #0b5394; font-size: medium;"> </span></span></p>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com1tag:blogger.com,1999:blog-3578987301814134971.post-23519919674126391402022-01-11T21:49:00.002+05:302022-02-23T00:06:57.068+05:30SATURN-V for Dummies Part-1: The History of Saturn-V<p></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; text-align: justify;"><span style="font-size: large;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; text-align: justify;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh9DcBEFcJ28Ti3PZcmdhmnl19qfffPdbcqdAFiCFz5niNXYJWjQkO6HvpQa8BHz3c9T-iwG5jO9n3QVlAIbk1-iVA3_I4lcoNXgBS1CbnSUgFnwt9JkthrrH1TAIP-fGnsvrxZPx2k2NLwyKU9dhyFw8DUkaWBb5uliE25y6xxvV60g6iev6KvHRHX=s4927" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3283" data-original-width="4927" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEh9DcBEFcJ28Ti3PZcmdhmnl19qfffPdbcqdAFiCFz5niNXYJWjQkO6HvpQa8BHz3c9T-iwG5jO9n3QVlAIbk1-iVA3_I4lcoNXgBS1CbnSUgFnwt9JkthrrH1TAIP-fGnsvrxZPx2k2NLwyKU9dhyFw8DUkaWBb5uliE25y6xxvV60g6iev6KvHRHX=s320" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif;"><span style="font-size: large;"><b>T</b></span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">he
mighty <b>Saturn-V</b> delivered humans to the <b>moon </b>in just about <b>3 days</b>. Standing at
a whopping height of <b>110m</b>, weighing nearly <b>3 million kilograms</b> on the launch
pad, and having a <b>payload capacity </b>of <b>140,000kg</b> to the <b>low earth orbit</b>, this beast continues
to be <b>the most powerful rocket </b>that mankind has ever built. But, why did
the rocket only function for a short span of only <b>6 years</b> (from <b>1967 to 1973</b>)
despite its exceptionally successful track record? How come a machine from the late
<b>1960s</b> stands unbeaten to this date? Answers to these questions (and many others)
can help us to answer why humans haven’t returned to the moon since <b>1972</b>.
Welcome to the series of articles named <b>“Saturn-V for Dummies”</b> in which we
gradually peel the Saturn-V to the core and get to see this beast with the lens
of <b>basic physics</b>. But first let’s go back in time, a time <b>long before Armstrong
set his first step on the moon</b>. Starting with the history of Saturn-V.</span></div><div class="separator" style="clear: both; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><br /></span></div></span></div>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;"><b>The History of Saturn-V: Or how did Saturn-V come into being?</b></span></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Saturn-V’s
ancestry can be traced back to the glorious <b>V2 rocket</b>. The German V2 rocket; developed by legendary <b>Wernher von Braun</b>, was meant to be used as a warhead
able to deploy payload (explosives) over a very large distance, it was <b>the world’s
first long-range missile</b> and was virtually unstoppable during the times of
<b>WWII</b>.<o:p></o:p></span></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjcyPSzvBaXBpmWFItZ0BB_B2L6yB63v7uYtADh0hIMdDD4anT9hf1Ajvu51y2sAdzeddKsbd8MaUaRrU7uVoNquROdGqK5XN4KhVRbboPiJZq7oEEN7ImYXc8hfKNI-a2SKZCw2qPbZYK6ct6zByfOOlFaVm2sSLQuxMxpC8i8ikVcVQw_4Dy_DHni=s1500" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1500" data-original-width="1000" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjcyPSzvBaXBpmWFItZ0BB_B2L6yB63v7uYtADh0hIMdDD4anT9hf1Ajvu51y2sAdzeddKsbd8MaUaRrU7uVoNquROdGqK5XN4KhVRbboPiJZq7oEEN7ImYXc8hfKNI-a2SKZCw2qPbZYK6ct6zByfOOlFaVm2sSLQuxMxpC8i8ikVcVQw_4Dy_DHni=s320" width="213" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">The V2 rocket<br /><br /></td></tr></tbody></table><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><div style="text-align: justify;"><span style="color: #0b5394; font-size: 13.5pt;">Apart
from the fact that V2 was a <b>formidable weapon</b> for the </span><b style="color: #0b5394; font-size: 13.5pt;">allies</b><span style="color: #0b5394; font-size: 13.5pt;">, it was a great
leap in the </span><b style="color: #0b5394; font-size: 13.5pt;">rocket technology</b><span style="color: #0b5394; font-size: 13.5pt;"> of the </span><b style="color: #0b5394; font-size: 13.5pt;">20th century</b><span style="color: #0b5394; font-size: 13.5pt;">, it is after all the first rocket ever to <b>reach space</b> by crossing the <b>Karman Line </b>(</span><b style="color: #0b5394; font-size: 13.5pt;">100km</b><span style="color: #0b5394; font-size: 13.5pt;"> altitude from sea level).</span></div></span><p></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">After
the win of allies over the <b>axis</b>, the talented <b>German aerospace workforce</b> under
<b>Nazi Germany</b> got divided into the <b>USA </b>and the former <b>USSR</b>. Von Braun was one of the engineers
who came to the US and thereby began the <b>golden age of rocket development</b> in
the States. It all started with American engineers <b>reverse engineering</b> and
understanding the V2 with the help of <b>German engineers</b>.<o:p></o:p></span></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEguryOVaqOUnAFTKURUVruJqrSQKjmRUVu_wxxNfC26qpYP6ufWFXcWIxL-R2E5Ybjuz3btH302zvGlMWqV9eyquOPEEQTcc7Kiv79XcviwfDFdM478yBM2WBo8OUYh2ZHofW4-9CEKAHq4ElrjSxbXCbxnPkVfYT7AtJum6HrE_cP9Z17JUM_smxGP=s3000" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-1: The History of Saturn-V The Dynamic Frequency" border="0" data-original-height="3000" data-original-width="2415" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEguryOVaqOUnAFTKURUVruJqrSQKjmRUVu_wxxNfC26qpYP6ufWFXcWIxL-R2E5Ybjuz3btH302zvGlMWqV9eyquOPEEQTcc7Kiv79XcviwfDFdM478yBM2WBo8OUYh2ZHofW4-9CEKAHq4ElrjSxbXCbxnPkVfYT7AtJum6HrE_cP9Z17JUM_smxGP=w258-h320" title="SATURN-V for Dummies Part-1: The History of Saturn-V" width="258" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Wernher von Braun with Saturn-V</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></div><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><b>The
Saturn series </b>of rockets (to which Saturn-V belongs) was basically the
successor of the <b>Jupiter series of rockets</b>. <b>The Juno rocket</b> which launched the
<b>first American satellite</b> to orbit belonged to the Jupiter series. Wernher von
Braun considered the Jupiter series to be a <b>prototype</b> and referred to it as “<b>an
infant Saturn</b>”.</span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjEtEobJntSDg21yXoJZy3tt4oO4asvjC76UAsXpjgUOBOW87xo1EVAN8ypXLlglLUDnV54ZIZFtuGO6RaHnuK1j8dDkJi-ctp3ejdaUn5UVKeuquTPgq8lxO6_gIuRo-NeM8cshA_l06Ah40lWS6bwttCTaMmjHboYBxw2CeS6RToE3iMN54AiyMvP=s3000" style="margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-1: The History of Saturn-V" border="0" data-original-height="3000" data-original-width="2045" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjEtEobJntSDg21yXoJZy3tt4oO4asvjC76UAsXpjgUOBOW87xo1EVAN8ypXLlglLUDnV54ZIZFtuGO6RaHnuK1j8dDkJi-ctp3ejdaUn5UVKeuquTPgq8lxO6_gIuRo-NeM8cshA_l06Ah40lWS6bwttCTaMmjHboYBxw2CeS6RToE3iMN54AiyMvP=w218-h320" title="SATURN-V for Dummies Part-1: The History of Saturn-V" width="218" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Juno-II rocket (Jupiter series)</td></tr></tbody></table><p></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Between
<b>1960-62</b>, <b>Marshall Space Flight Centre (MSFC)</b> designed a series of Saturn
rockets each with a varied capability ranging from <b>earth orbits</b> to <b>lunar
missions</b>. Originally the rockets followed a <b>‘C’</b> nomenclature i.e., rockets were
named as <b>C-1, C-2,</b> etc.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b>NASA</b>
initially planned to use <b>C-3 </b>for <b>Earth Orbit Rendezvous (EOR)</b> which needed <b>4-5
launches per lunar mission</b>. However, MSFC planned for an even bigger rocket,
the <b>C-4</b>, which would use <b>four F-1 engines</b> (will be covered in detail in the 3<sup>rd</sup> article)
in its first stage, an enlarged C-3 second stage, and a couple more
upgrades. The C-4 could carry out the EOR Lunar mission in just <b>2 launches</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: #0b5394;"><span style="font-size: 13.5pt;">On
</span><b><span style="font-size: 13.5pt;">January 10th,</span><span style="font-size: 15px;"> </span><span style="font-size: 13.5pt;">1962</span></b><span style="font-size: 13.5pt;">, NASA announced planned for a </span><b style="font-size: 13.5pt;">C-5 launch
vehicle</b><span style="font-size: 13.5pt;"> with a </span><b style="font-size: 13.5pt;">41000kg </b><span style="font-size: 13.5pt;">payload capacity to the moon. The first rockets in the
series were </span><b style="font-size: 13.5pt;">Saturn-1</b><span style="font-size: 13.5pt;"> originally named </span><b style="font-size: 13.5pt;">C-1</b><span style="font-size: 13.5pt;"> and </span><b style="font-size: 13.5pt;">Saturn-1(b) / C-1(b)</b><span style="font-size: 13.5pt;">.
Both Saturn-1 and 1(b) served as proof of concept and helped collect
critical flight data for the upcoming Saturn-V/ C-5. With NASA's validation of
C-5 as the </span><b style="font-size: 13.5pt;">official launch vehicle</b><span style="font-size: 13.5pt;"> for the </span><b style="font-size: 13.5pt;">Apollo program</b><span style="font-size: 13.5pt;">, the 'C' was dropped
from the name, and 'Saturn' was added.</span></span></span></p><p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><div style="text-align: left;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjrMg7SqeX2mn8OUZ4hKCxzH7mcmsgroa329JUsXaqd1205nC7BP_NzLwUa7hxoTDMB7YXIs6AryPPa0W08_d97VLWVjHLg9PhTHF4fS-Hi0pcrdJI8-Qy3sq4rfYEJxnUnanqN8unwDs-g6bDmb0ph3wYNS1mxHsHDOOwRyAx7dARYU6XXhwDbY6uL=s1024" style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-1: The History of Saturn-V" border="0" data-original-height="1024" data-original-width="776" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEjrMg7SqeX2mn8OUZ4hKCxzH7mcmsgroa329JUsXaqd1205nC7BP_NzLwUa7hxoTDMB7YXIs6AryPPa0W08_d97VLWVjHLg9PhTHF4fS-Hi0pcrdJI8-Qy3sq4rfYEJxnUnanqN8unwDs-g6bDmb0ph3wYNS1mxHsHDOOwRyAx7dARYU6XXhwDbY6uL=w243-h320" title="SATURN-V for Dummies Part-1: The History of Saturn-V" width="243" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Saturn-1</td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhZg2MMnLWs3YjYT9ADGr79FdZbd0oqC18MRkX4F8gLkmT6tTlKgKClOzzwL2e6iJeuTxh3DI9Tzc-MSFvYs4RgL-_jFEo7P5eZC0H4o_mL7J0Y3AQ4pCv4qGXtaqF5cCNqO7_jzmHSlMn5L9F_-cm2OtiCsCPbUKwVXmV24q8yOxQIf09OfndHceQa=s1600" style="font-family: "Footlight MT Light", serif; font-size: 18px; margin-left: auto; margin-right: auto;"><img alt="SATURN-V for Dummies Part-1: The History of Saturn-V" border="0" data-original-height="1600" data-original-width="1095" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhZg2MMnLWs3YjYT9ADGr79FdZbd0oqC18MRkX4F8gLkmT6tTlKgKClOzzwL2e6iJeuTxh3DI9Tzc-MSFvYs4RgL-_jFEo7P5eZC0H4o_mL7J0Y3AQ4pCv4qGXtaqF5cCNqO7_jzmHSlMn5L9F_-cm2OtiCsCPbUKwVXmV24q8yOxQIf09OfndHceQa=w219-h320" title="SATURN-V for Dummies Part-1: The History of Saturn-V" width="219" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Saturn-1b</td></tr></tbody></table><div style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><br /></span></div><div style="text-align: justify;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">By
</span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">1962</b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">, NASA finalized to proceed with von Braun’s designs, as a result, the Apollo
program <b>accelerated faster</b>. Apart from selecting the <b>rocket design</b>, NASA had to face
several other issues regarding the <b>nitty-gritty details</b> of the rocket, which were eventually resolved.</span></div></div></div>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">But
what actually was this Saturn-V machine? What did it constitute? What were the points of the major issue? There are more <b>n-number</b> of questions waiting to be answered. If
you have such questions stuck inside your mind then stay tuned till our next
article in the series in which we will be <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/saturnV-for-dummies-pt2-machine-overview.html" target="_blank">stripping down the machine to unveil the extra-extraordinary engineering</a></b> that went behind building this beast. Once it is published, the link will be available here.<o:p></o:p></span></span></p><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "footlight mt light", "serif"; font-size: 13.5pt; line-height: 19.26px; outline: 0px; padding: 0px; vertical-align: baseline;">If you like our articles then please <b style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">follow this blog</b> to get the <b style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">latest updates directly to you</b> <o:p></o:p></span><b style="background: 0px 0px; border: 0px; font-family: "footlight mt light", serif; font-size: 18px; outline: 0px; padding: 0px; vertical-align: baseline;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span></div><div class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><br /></div><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"><b style="background: 0px 0px; border: 0px; color: #474747; outline: 0px; padding: 0px; vertical-align: baseline;">To know the basics of the quantum world, astronomy and space exploration you can check out the book <a href="https://www.amazon.in/dp/B07TBSP6JT" style="background: 0px 0px; border: 0px; color: #e94bc3; outline: 0px; padding: 0px; text-decoration-line: none; transition: color 0.17s ease 0s; vertical-align: baseline;" target="_blank">"Through the wormhole" on amazon kindle</a> by Ratnadeep Das Choudhury.</b></span></span></div></div><div style="background: 0px 0px rgb(255, 255, 255); border: 0px; color: red; font-family: Lato, sans-serif; font-size: 15px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"></div><div class="MsoNormal" style="background: 0px 0px white; border: 0px; 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padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span><span style="background: 0px 0px; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"> </span>-Ayushman Dash</span></div></div><div style="background: 0px 0px; border: 0px; color: black; font-family: "times new roman"; font-size: medium; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;"><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background: 0px 0px; border: 0px; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; outline: 0px; padding: 0px; text-align: left; vertical-align: baseline;"> Chief Space Exploration Writer</span></div><div style="background: 0px 0px; border: 0px; margin: 0px; outline: 0px; padding: 0px; text-align: right; vertical-align: baseline;"><span style="background-color: initial; color: lime; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">The Dynamic Frequency</span></div></div></span></div></div></div>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"> <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%;"><o:p><span style="color: #0b5394;"> </span></o:p></span></p><br /><p></p>Ayushman Dashhttp://www.blogger.com/profile/16765055881845472552noreply@blogger.com10tag:blogger.com,1999:blog-3578987301814134971.post-24555186441381992082022-01-03T22:37:00.002+05:302022-01-03T23:41:47.896+05:30The Insane Engineering of SpaceX’s Merlin Engines Part-2: Introduction to Turbopump<p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEi4PkKrEjcsn7Oy0YDWyPOivvYhDAdf1E5jsBcvQWkypy7aYZdbyl8BFdZZN-k1tVOTFHm7zfvHUuYH6HjQCFgOe_9ET2A2XJcnWRwCQGKpKwgUiA-z1RRi7CHJ-EsZcO2zZ3sLVn_VH3RTmmBUg6da7o_i8LKDo_t5S5jizXmKm7U9TahpjaINq8NeEQ=s4752" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3168" data-original-width="4752" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEi4PkKrEjcsn7Oy0YDWyPOivvYhDAdf1E5jsBcvQWkypy7aYZdbyl8BFdZZN-k1tVOTFHm7zfvHUuYH6HjQCFgOe_9ET2A2XJcnWRwCQGKpKwgUiA-z1RRi7CHJ-EsZcO2zZ3sLVn_VH3RTmmBUg6da7o_i8LKDo_t5S5jizXmKm7U9TahpjaINq8NeEQ=s320" width="320" /></a></span></div><p></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">First of all, I wish all my beloved readers a very happy and prosperous new year 2022. I wish in this new year, we all can learn many new things and can apply those for the betterment of mankind.</span></span></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Before reading this
article, I will highly recommend you to read the <b><a href="https://thedynamicfrequency.blogspot.com/2021/11/spaceX-merlin-introduction.html" target="_blank">part-1 article in this series of Insane engineering of SpaceX’s Merlin Engine</a></b> for better understanding. In
this article, I will be introducing you to the basic concept of Turbopumps used
in cryogenic engines. In later articles, we can use this knowledge for
understanding Merlin’s design and functionality.</span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, let’s start…<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now you have a <b>fuel
tank</b> and <b>propellants</b>. But these two things alone aren’t enough for the
effective combustion in the combustion chamber to create a thrust greater than
the gravity and ultimately lift up the rocket. So, to solve this issue rocket
engineers developed a device named<b> turbopump.</b> Turbopump <b>helps to send the
propellant to the combustion chamber at a high speed and effectively. </b>The Merlin engines use LOX and RP-1 turbopump. It delivers about 10,000 horsepower or 7500 kilowatts of power with 36,000 revolutions per minute.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, the obvious
question may arise that, <b>why we don’t use the normal electric pump that we use in
our home as it is easier and cheaper to manufacture?</b><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The answer is, <b>we don’t use them as
they would be bulky and also need a large amount of power supply in form of
batteries which in turn also increase the overall weight</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, now that you know
the importance of turbopump, then let’s dig more into its technical aspects.
<b>Turbopump mainly has two main components, a rotodynamic pump and a gas turbine.</b>
In simple terms, <b>a rotodynamic pump is a machine to pump fluid (here,
propellent) by using a rotating impeller. </b><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, both the above
components are mounted together. Since the whole system has a gas turbine, so
we need gas to drive it. To achieve this, <b>we have to circulate the liquid
hydrogen around the hot nozzle and combustion chamber of the rocket and pass
the resulting gaseous hydrogen to the turbine to drive it</b>. This also helps to
cool the nozzle and combustion chamber whose temperature can reach up to <b>3000
degrees Celsius</b> and <b>cause material damage</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">After the gas did its
job of driving the turbine, the <b>gas is again sent back to the combustion chamber</b>.
The rotating turbine then drives the <b>impeller which helps to increase the flow
of liquid hydrogen or oxygen</b>. This whole configuration is called an <b>expander
cycle</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">However, even after
using this configuration, <b>we cannot pump the liquid hydrogen with the high speed
required to sustain enough trust for the effective thrust</b>. This is because of the
simple property of gas. As we all know the more we heat a substance more it
expands and has greater kinetic energy. So, <b>here to heat the liquid hydrogen
we use the thermal energy radiated from the hot nozzle and combustion chamber</b>.
But the problem is that that heat is also not sufficient enough to drive the
turbine at the required speed to pump the liquid hydrogen for effective thrust.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, to tackle this
problem, engineers have developed something known as <b>Gas Generator Cycle</b>. In
this method, <b>a small portion of the liquid hydrogen is burned in a</b> <b>small
separated combustion chamber</b> and then the resulting hot gases are used to drive
the gas turbine. The turbine then turns the impeller and it ensures high speed
of liquid propellent flow around the nozzle and ultimately to the main
combustion chamber of the engine. <o:p></o:p></span></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg7F9wcPfsQQcxPJIHvK_ZoDqmZ0rVuDwHi1pqSqyVezLr-qgENdwAdT_1BD9gcSATcNbcJxLWjk1S_Q0D-3g7It4ipBnW3bevMo6SGv9n4DWcYUOgl0Qc7MeqjDHb273P6RgGdyyBloxucaUTTffMY82WMAYK4zqSwBK-qGFwa_82jySnl7KhMbek8vg=s675" style="margin-left: auto; margin-right: auto;"><img alt="Gas-Generator Cycle The Insane Engineering of SpaceX’s Merlin Engines Part-2: Introduction to Turbopump" border="0" data-original-height="675" data-original-width="550" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEg7F9wcPfsQQcxPJIHvK_ZoDqmZ0rVuDwHi1pqSqyVezLr-qgENdwAdT_1BD9gcSATcNbcJxLWjk1S_Q0D-3g7It4ipBnW3bevMo6SGv9n4DWcYUOgl0Qc7MeqjDHb273P6RgGdyyBloxucaUTTffMY82WMAYK4zqSwBK-qGFwa_82jySnl7KhMbek8vg=w261-h320" title="Gas-Generator Cycle The Insane Engineering of SpaceX’s Merlin Engines Part-2: Introduction to Turbopump" width="261" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Gas-Generator Cycle</td></tr></tbody></table>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>But this design is
inefficient as a small amount of exhaust gas is totally lost and results in
loss of potential fuel</b>. Although there are various ways to increase the efficiency of the system like diverting the hot gas due to the hot nozzle to the
small combustion chamber and also diverting the exhaust which was getting lost
earlier to the main combustion chamber of the engine. This results in a half-burned
fuel-rich mixture that has both high temperature and pressure.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Also, one must remember that in this article I used liquid hydrogen (LH2) and oxygen (LOH) to explain to you the working of turbopumps, which is traditionally being used in various engines. But, in merlin engines, a combination of </span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 18px;">LOX and RP-1 is used.</span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The main challenge
with any cryogenic engine is to maintain specific oxygen to hydrogen ratio
which is also known as <b>the propellant mixture ratio</b>. This crucial job is totally
dependent on the effective design of the turbopump. Also, <b>one of the main
challenges of the turbopump design is that in turbopumps the propellant flow rate
is controlled by a gas turbine which is also gets controlled by the propellant
combustion or hot propellant gas</b>. If this design is achieved successfully then
one can finally manufacture a cryogenic engine. But in reality, it's not the
case. Only a handful of countries successfully manufactured and used this type
of engine.</span><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">In my next article, I will try to introduce you all to the different versions of Merlin engines with the specifications and problems of each of them. </span></span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 18px;">Once it is published, the link will be available in this article.</b></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>follow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know the basics of the quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><span style="color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 19.26px; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; font-family: lato, sans-serif; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i style="color: #0b5394; font-family: times, "times new roman", serif;">To know my other works visit my site - <a href="https://ratnadeepdchoudhury.com/" rel="" target="_blank"><b>ratnadeepdchoudhury.com</b></a></i></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; 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font-style: normal; margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>I also have a youtube channel where I make videos on Productivity & Growth, Physics and Technology. Check it now on <b><a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank">YouTube</a></b></i></span></div></i></span></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span></div><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><div style="color: black; font-family: "Times New Roman"; font-size: medium;"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span>-Ratnadeep Das Choudhury</span></div></div><div style="color: black; font-family: "times new roman"; font-size: medium; text-align: justify;"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-67931703891660708602021-11-17T00:19:00.003+05:302022-01-03T22:39:10.232+05:30The Insane Engineering of SpaceX’s Merlin Engines Part-1: The Introduction<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj5gDT1Cftby0CFJC9rJADT_aSE76WITwUoGZ-Au8q7sae6NG3U6ycpQKg1kgioYzesdfVyZgKoW1Xzwo_Kfvj_hPO0WjGTk_jSfveGwZgI6Z7rWML00jxC-b6g2HGaRftot-gkfulWSRHWMED25st-y6r-qGjGg_SFxg4CCDEs68fab4r9GYUm_w2GZw=s1280" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="853" data-original-width="1280" height="213" src="https://blogger.googleusercontent.com/img/a/AVvXsEj5gDT1Cftby0CFJC9rJADT_aSE76WITwUoGZ-Au8q7sae6NG3U6ycpQKg1kgioYzesdfVyZgKoW1Xzwo_Kfvj_hPO0WjGTk_jSfveGwZgI6Z7rWML00jxC-b6g2HGaRftot-gkfulWSRHWMED25st-y6r-qGjGg_SFxg4CCDEs68fab4r9GYUm_w2GZw=s320" width="320" /></a></div><p></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">In today’s aerospace
world<b>, SpaceX needs no introduction. Being the largest private space
technological company</b>, <b>SpaceX</b> proved everyone wrong. But, to make this happen
in the first place, they need a solid and reliable engine for their rocket’s
and here they introduced their <b>Merlin Engines</b>.</span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Known for their sheer
power and reliability along with the reusable nature of the rockets, the
SpaceX’s brought the <b>launching cost dramatically lower than previous
generations of rockets</b>.</span><o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhGmybfLusvimx72Pjmg5VmBHsuVSQrcSp01sJRKXO5FYDBDyperGxFyDv_5W4vZP1z9uRYKwXmYZ4G-u-vPey1-Ghl2rfFFegB1WyIqoOS3Y0WqhBQDcnZmRXI2-ofXxN9dtP7V5sosThYv7XRYWcL0yJCYeD0SUWaUsrVfooga7N3i-NykYPPMtzrgQ=s600" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="600" data-original-width="400" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhGmybfLusvimx72Pjmg5VmBHsuVSQrcSp01sJRKXO5FYDBDyperGxFyDv_5W4vZP1z9uRYKwXmYZ4G-u-vPey1-Ghl2rfFFegB1WyIqoOS3Y0WqhBQDcnZmRXI2-ofXxN9dtP7V5sosThYv7XRYWcL0yJCYeD0SUWaUsrVfooga7N3i-NykYPPMtzrgQ=s320" width="213" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Elon Musk - Founder, CEO and chief engineer of SpaceX</td></tr></tbody></table><p></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">To introduce you to
the insane engineering they used for developing the Merlin engines, I decided
to make a series of articles through which I will explain to you the mechanism
behind it in a very simple way. </span><o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhYYjVNK-ZtPSCYDdKn0yknlNBWF5uiKXlPz3hK17NNZje3XbotDCkVgwgKFOogU8rBxSagp6Pvp_av3TGU_wniQvDy9sx5eMtTugr8CQoQQxpOycRDmaaSzmm07nfD9T42wkaXMUrQYCTZ9zTDLCzXilBNoActC3eZall6DroXfWBrzNMNZfcZwcAMSg=s1024" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1024" data-original-width="769" height="320" src="https://blogger.googleusercontent.com/img/a/AVvXsEhYYjVNK-ZtPSCYDdKn0yknlNBWF5uiKXlPz3hK17NNZje3XbotDCkVgwgKFOogU8rBxSagp6Pvp_av3TGU_wniQvDy9sx5eMtTugr8CQoQQxpOycRDmaaSzmm07nfD9T42wkaXMUrQYCTZ9zTDLCzXilBNoActC3eZall6DroXfWBrzNMNZfcZwcAMSg=s320" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Merlin Engine</td></tr></tbody></table><p></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">To understand the
mechanism, you need to know some technical terms and the working of the really
important parts. So, let’s start from that.</span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red; font-size: x-large;"><b>Important
Terminologies</b></span><o:p style="font-size: 13.5pt;"></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Before starting, I
want to introduce some of the important terminologies which are very important
in order to understand the rocket science behind these magnificent machines.
<b>Remember, that these components are used in various applications in mechanical
engineering, but here, I am only going to explain to you about their usage in
aerospace engineering.</b> So, let’s start.</span><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b><span style="color: #6aa84f;">1) Cryogenics</span></b><o:p></o:p></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Cryogenics is the
branch to study the <b>production and behaviour of materials at a very low
temperature</b>. So, a cryogenic engine means an engine that uses cryogenic tech for
its functionality. Now, here I oversimplified the word cryogenic tech. So,
let’s go a little bit deeper. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Here, cryogenic tech
includes the cryogenic fuel and materials & engineering to handle them. To
propel rockets we need efficient fuels. The most preferred ones are oxygen and
hydrogen. Why???.... We will discuss it later. For now, just remember these
fuels.<o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjRgx86kbSbkAhoMhRcTSK8G_ZvE5jWO6yV0Jfju1QvprKzcVMPS2hM4KP0KD-TWwmy4aYUPW2kmEcK1Gz25S_7HSKDrU29Jb3vjkMle_nUFzrjHDSLVKMa8uYbVxTVZ9QsgsmrM7d9gJUfSwR0jGIS_uyiaIuZdpckitgxWsNT_0FyJBZBBVcJILn9Aw=s602" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="422" data-original-width="602" height="224" src="https://blogger.googleusercontent.com/img/a/AVvXsEjRgx86kbSbkAhoMhRcTSK8G_ZvE5jWO6yV0Jfju1QvprKzcVMPS2hM4KP0KD-TWwmy4aYUPW2kmEcK1Gz25S_7HSKDrU29Jb3vjkMle_nUFzrjHDSLVKMa8uYbVxTVZ9QsgsmrM7d9gJUfSwR0jGIS_uyiaIuZdpckitgxWsNT_0FyJBZBBVcJILn9Aw=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Liquid Hydrogen</td></tr></tbody></table><p></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">The problem with Oxygen
and Hydrogen is that they are in the gaseous state at room temperature. Since they
are in a gaseous state, <b>so we cannot store enough quantity of oxygen and
hydrogen in a compact fuel tank for the whole journey of the rocket</b>. To solve
this problem, we use the knowledge of cryogenics. <b>We can store them in liquid form and to do so we have to cool them down and store them in a
pressured tank</b>. Along with this, the materials used to manufacture these tanks
have to specialise for this purpose. <o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgn_74b0CieMIFM95pwgn9qnWC3f4Z3lQ_X_iXWIycpUmi2OtxrUSHEp7D2PfJzdlrUzlscd1dGgAD_HcLEZ5iJN1Uj7bYjAyJ6LuevByVv61d6tLBDdcxnp5a4AJF3P15weeuC2DuKwGPHGlWbR3dJTsjzMtoz4yqQv6LxfmTARI1AIMFV3_xbFbEd3w=s657" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="599" data-original-width="657" height="292" src="https://blogger.googleusercontent.com/img/a/AVvXsEgn_74b0CieMIFM95pwgn9qnWC3f4Z3lQ_X_iXWIycpUmi2OtxrUSHEp7D2PfJzdlrUzlscd1dGgAD_HcLEZ5iJN1Uj7bYjAyJ6LuevByVv61d6tLBDdcxnp5a4AJF3P15weeuC2DuKwGPHGlWbR3dJTsjzMtoz4yqQv6LxfmTARI1AIMFV3_xbFbEd3w=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Liquid Oxygen storage facility</td></tr></tbody></table><p></p><p class="MsoNormal"><b style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #6aa84f;">2) Propellent</span></b></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">In simple terms, the propellant is a substance that propels a vehicle in a particular direction. As
we have already discussed in the previous point about <b>liquid hydrogen (LH2)</b> and
<b>liquid oxygen (LOX)</b>, they are the fuel or in technical terms the propellant of
the rocket. In LH2 and LOX combination, <b>the LH2 is called the rocket
fuel component and LOX is known as the oxidizer</b>.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">One must note that
<b>Merlin engines don’t use liquid hydrogen and liquid oxygen combination for the
fuel rather they use the combination of liquid oxygen (as oxidizer) and RP-1(as
rocket fuel) as propellants</b>. <b>RP-1 stands for Rocket Propellent-1 or Refined
Petroleum-1</b>. This is a highly refined version of kerosene which is primarily used
as rocket fuel. <b>Its biggest advantage is that it's cheaper and stable at room
temperature as compared to liquid hydrogen. But at the same time, it has also a lower specific impulse (see point 6 for details) than liquid hydrogen</b>. These all characteristics made RP-1
a preferable fuel for merlin engines and Falcon rockets as their main aim is
to reduce the cost of each launch even while maintaining high efficiency.<o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiL4ZG6SnJC2CZhQR5oNnwtSZRJ0gI0eiOLEO7Fb4lx6v0KavPdUrT2CPhWLG96sCitPPV1ic6v9G5xOakRiVYROMJvWOi_rqSJOmUTbZzPOYT6znJ0pQWq2ne604wcV8MK8NAg7sd8YiJ2jJj54aYCPT6y-cbAneSip8OY3RKxEe5og6WcmkLDVnrxUA=s2048" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1245" data-original-width="2048" height="195" src="https://blogger.googleusercontent.com/img/a/AVvXsEiL4ZG6SnJC2CZhQR5oNnwtSZRJ0gI0eiOLEO7Fb4lx6v0KavPdUrT2CPhWLG96sCitPPV1ic6v9G5xOakRiVYROMJvWOi_rqSJOmUTbZzPOYT6znJ0pQWq2ne604wcV8MK8NAg7sd8YiJ2jJj54aYCPT6y-cbAneSip8OY3RKxEe5og6WcmkLDVnrxUA=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Liquid Hydrogen and RP-1 storage facility</td></tr></tbody></table><p></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #6aa84f;"><b>3) Impeller</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">The impeller is a very
essential component in any rocket, <b>especially in turbopumps which I will cover
in my next article</b>. For now, I will try to explain it in a very simple way. In
mechanical engineering, an impeller is a component that rotates to increase
the pressure and thus control the flow of a fluid. Here, in our case the
fluid is propellent. <b>Impellers used in cryogenic engines like merlin have a series of curved vanes fitted inside shroud plates. The whole
structure is rounded. The middle portion of the impeller where the blades are
connected together is called the eye.</b><o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEg4ZU7HgCa8mGtRiG2GY47zafQD_Gyis96Iotl44kHZRyqwnCwP40FEfU2Pz0B6icdVD0a5fl44o0ZKsRjeQZ2KGF49UtyhxuPqeHzvkei06GqeLvJO_pSUk4PmLGMt2EkxDu6BmFdeAO0zYXJ5q4NxV74FqpnoQMja-00bJfI9JQ8Huhzc_zyxcwvHgw=s608" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="291" data-original-width="608" height="153" src="https://blogger.googleusercontent.com/img/a/AVvXsEg4ZU7HgCa8mGtRiG2GY47zafQD_Gyis96Iotl44kHZRyqwnCwP40FEfU2Pz0B6icdVD0a5fl44o0ZKsRjeQZ2KGF49UtyhxuPqeHzvkei06GqeLvJO_pSUk4PmLGMt2EkxDu6BmFdeAO0zYXJ5q4NxV74FqpnoQMja-00bJfI9JQ8Huhzc_zyxcwvHgw=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Impellers</td></tr></tbody></table><p></p><p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">When the impeller rotates,
the liquid propellent surrounding the rocket also rotates.<b> Due to curved vanes
and rotation of the impeller, the propellent moves radially out due to centrifugal
force and thus increasing the pressure & kinetic energy of the propellant
as the rotational mechanical energy is transferred from the impeller to the
propellent</b>. At the eye of the impeller, since the propellent get displaced, so a
negative pressure is created and thus helping the impeller to suck in more
propellent from the tank.</span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b><span style="color: #6aa84f;">4) Pintle Injector</span></b><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">In a cryogenic rocket
engine, if you only have propellants, effective combustion cannot take place
for effective propulsion as it will be very difficult to mix those propellents. Here, the pintle injector comes into the picture. <b>It helps to
ensure appropriate intermixing of the propellants. It consists of two
concentric tubes and one opening. Once the propellants get mixed it goes
through atomization, then it goes to the main combustion chamber.</b><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #6aa84f;"><b>5) Pyrophoric Ignition</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">To understand this,
you first have to understand the meaning of pyrophoricity. A pyrophoric<b> substance is a substance that ignites instantly once it gets in contact with
oxygen</b>. So, <b>pyrophoric ignition means the ignition in the combustion chamber by
using pyrophoric material. SpaceX merlin uses TEA-TEB pyrophoric ignition
method. TEA stands for Triethylaluminium and TEB stands for Triethylboron</b>. Both
are colourless pyrophoric liquids.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b><span style="color: #6aa84f;">6) Specific Impulse </span></b><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="font-size: 13.5pt;">In simple terms, specific impulse means the thrust which has produced by a propellent per unit
rate of the same. Its usually represented by </span><b><span style="font-size: 13.5pt;">I</span><sub><span style="font-size: x-small;">SP</span></sub></b><span style="font-size: 13.5pt;">. </span><o:p style="font-size: 13.5pt;"></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #6aa84f;"><b>7) Calorific Value </b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Calorific value is the
amount of energy actually stored in a certain kind of fuel or propellant. It
can be measured by measuring the total amount of heat energy is produced after
its complete combustion. Its unit is Joules per Kilogram.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b><span style="color: #6aa84f;">8) Thrust Vector
Control (TVC)</span></b><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Controlling a rocket is
a very difficult task, especially when you know that those rockets are made for
carrying out operations in space. <b>The problem is that in space, the density of
air is either thin or null. So, the control methods we are familiar with using
in aeroplanes fail to execute and do their job in space. Here, the technology
of thrust vector control comes into the picture. In this technique, the
direction, altitude, or angular velocity of rockets can be controlled by the engine's thrust. </b>Its primarily achieved by controlling the
direction of the rocket’s nozzle. There are many other complex methods to
achieve this capability, which I am not going to discuss here as it is out of this article's scope.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #6aa84f;"><b>9) Converging-Diverging
(CD) Nozzle</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">As the name revers, this
type of nozzles has a converging and diverging shape. It was first invented by a Swedish
engineer named<b> Gustaf De Laval</b> and that’s why this type of nozzle is also
known as the <b>de Laval nozzle</b>. This type of nozzle was first used in rockets by
<b>Robert Goddard</b>. CD nozzle’s<b> primary job is to accelerate the gas passing
through it to supersonic speed by converting the heat energy to kinetic energy
and pressurising the gas, all due to its shape</b>. <o:p></o:p></span></p><p class="MsoNormal"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiRwSWUvO1h_z8R0a2yAdX_s94PsgQFcVevItob4o5Y1cIVqybrLWl-w1qURXaAmqTLSPR1z10HlFqADj88ZZGKKoHgeV06gcm8zZ9GdD9bNivqqhHfZ4qtjZIgysoQUu018UOimWJdbGP9DRQjFOmHumiArirwNLai5zUcF05HaLBBlQ1AUan-ShrMvg=s2048" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1707" data-original-width="2048" height="267" src="https://blogger.googleusercontent.com/img/a/AVvXsEiRwSWUvO1h_z8R0a2yAdX_s94PsgQFcVevItob4o5Y1cIVqybrLWl-w1qURXaAmqTLSPR1z10HlFqADj88ZZGKKoHgeV06gcm8zZ9GdD9bNivqqhHfZ4qtjZIgysoQUu018UOimWJdbGP9DRQjFOmHumiArirwNLai5zUcF05HaLBBlQ1AUan-ShrMvg=s320" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Nine CD nozzles configuration at the base of Falcon-9 rocket </td></tr></tbody></table><p></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Although there are a
hell lot of components which is needed to be discussed before moving forward, for now, this much knowledge is enough to understand the <b><a href="https://thedynamicfrequency.blogspot.com/2022/01/spaceX-merlin-intro-to-turbopump.html" target="_blank">basics of turbopump</a></b>, which is the topic for our next article.</span></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>follow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article on mobile).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><span style="color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 19.26px; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; font-family: lato, sans-serif; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i style="color: #0b5394; font-family: times, "times new roman", serif;">To know my other works visit my site - <a href="https://ratnadeepdchoudhury.com/" rel="" target="_blank"><b>ratnadeepdchoudhury.com</b></a></i></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; text-align: left; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394; font-size: 11pt; line-height: 15.6933px;">To meet more physics Enthusiasts please join our</span><span style="font-size: 11pt; line-height: 15.6933px;"> </span></i><span style="font-size: 11pt; line-height: 15.6933px;"><a href="https://www.facebook.com/The-Dynamic-Frequency-452248138674077/?modal=admin_todo_tour"><b><i>Facebook Page</i></b><i> </i></a></span></span></div></div><div style="text-align: left;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><span style="color: #0b5394;">Also for the latest updates of my posts join me on </span><b><a href="https://twitter.com/TheDynamicFreq1">Twitter</a></b></i><span style="color: #0b5394; font-size: 13.5pt;"></span></span></div><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><div style="font-family: "Times New Roman"; font-style: normal; margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>I also have a youtube channel where I make videos on Productivity & Growth, Physics and Technology. Check it now on <b><a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank">YouTube</a></b></i></span></div></i></span></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-43469799984636987662021-07-28T23:27:00.003+05:302021-07-28T23:37:28.922+05:30Indian Space Station – An upcoming billion dollars project representing a dream of billions of people<p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYUyLfA6GtJl2wq3Tmn0m6z4SXXjsEpMzQNh2FwYW0Z9k5pQBtCkGsQ9VCXyvKcKZe8wzgZctfaFNjnSkcYoVrslAdXx1Rouy_E8Q1ZN4PaTTUSEVDpagsPvPtxyywDlW3pzdYz0bFlc6N/s567/earth-6013927__340.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="340" data-original-width="567" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYUyLfA6GtJl2wq3Tmn0m6z4SXXjsEpMzQNh2FwYW0Z9k5pQBtCkGsQ9VCXyvKcKZe8wzgZctfaFNjnSkcYoVrslAdXx1Rouy_E8Q1ZN4PaTTUSEVDpagsPvPtxyywDlW3pzdYz0bFlc6N/s320/earth-6013927__340.jpg" width="320" /></a></span></div><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><br /><span style="color: #0b5394;"><br /></span></span><p></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">International Space Station
– an iconic satellite-laboratory orbiting our earth with a speed of about <b>7.66
km/sec</b>, already inspired a whole new generation of future space scientists to
dream big. Although, it was not the first space station introduced to the
world. <b>Both USA’s Skylab and Soviet’s Mir, inspired to create ISS.</b></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">But this time, <b>India
also wants to join the elite league</b>. As already, India is preparing for its first
ever human space flight, ISRO’s chairman confirmed about <b>Indian Space Station</b>. It
will be undertaken after the success of <b><a href="https://thedynamicfrequency.blogspot.com/2019/06/indian-humanspace-program-isro-gslvmk3.html" target="_blank">Gaganyaan mission</a></b> that is, India’s
first human flight mission. The whole space station is expected to have a weight
of 20 tonnes. The spacecraft will be placed in <b>Low earth Orbit (300-400km)</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Indian space Station
will be a <b>modular space station like ISS</b>. This means different parts will be
launched from different launches as it is not possible to launch such a big
structure at once.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgONVY6JNZuFoK_N3XFocOu8S4B3qiWakpC0n3zdAGUStwB5kiWQCCoxNkA2f-VyP4EO8uAIMpFoNWVhlcz1AbEQzOZjOxfp5pKIMcAJuenzuJdCQB2Pkdb9AIuRQNl-cCkIeXSPrGjDyg4/s1200/e62c4da1c5f4503b14f5699268fd6fa2.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="900" data-original-width="1200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgONVY6JNZuFoK_N3XFocOu8S4B3qiWakpC0n3zdAGUStwB5kiWQCCoxNkA2f-VyP4EO8uAIMpFoNWVhlcz1AbEQzOZjOxfp5pKIMcAJuenzuJdCQB2Pkdb9AIuRQNl-cCkIeXSPrGjDyg4/s320/e62c4da1c5f4503b14f5699268fd6fa2.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">A model of Indian Space Station with docked Gaganyaan</td></tr></tbody></table><br /><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The habitat modules will
have their <b>own sets of solar panels for power generation</b>. As per the recent
announcements, there will be two habitat modules. Each of the, will also have
<b>their own radiators</b>. Radiators become more important in space stations, because
when they orbit, one side will be extremely hot and other side will be extremely
cold. So, radiators help to maintain a proper temperature balance inside to
maintain critical life support environment for the crew. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Along with these, the
habitat modules will have <b>their own docking ports so that spacecrafts like Gaganyaan
can dock with them</b>. If ISRO provides the international docking port facility in
this space station then, in future, other country’s spacecrafts can also dock
there.<b> The Habitat module is also expected to support 2-4 people</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">For now, everything is
dependent on Gaganyaan mission and its success will determine the timeline for
the completion of Indian Space Station. Wishing all the best to ISRO.</span><o:p></o:p></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing to our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article in mobile).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><span style="color: #0b5394; font-family: "footlight mt light", serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 19.26px; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; font-family: lato, sans-serif; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i style="color: #0b5394; font-family: times, "times new roman", serif;">To know my other works visit my site - <a href="https://ratnadeepdchoudhury.com/" rel="" target="_blank"><b>ratnadeepdchoudhury.com</b></a></i></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; text-align: left; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394; font-size: 11pt; line-height: 15.6933px;">To meet more physics Enthusiasts please join our</span><span style="font-size: 11pt; line-height: 15.6933px;"> </span></i><span style="font-size: 11pt; line-height: 15.6933px;"><a href="https://www.facebook.com/The-Dynamic-Frequency-452248138674077/?modal=admin_todo_tour"><b><i>Facebook Page</i></b><i> </i></a></span></span></div></div><div style="text-align: left;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><span style="color: #0b5394;">Also for the latest updates of my posts join me on </span><b><a href="https://twitter.com/TheDynamicFreq1">Twitter</a></b></i><span style="color: #0b5394; font-size: 13.5pt;"></span></span></div><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><div style="font-family: "Times New Roman"; font-style: normal; margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>I also have a youtube channel where I make videos on Productivity & Growth, Physics and Technology. Check it now on <b><a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank">YouTube</a></b></i></span></div></i></span></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-42180002108523046482021-07-06T23:35:00.002+05:302021-07-06T23:35:15.874+05:30Speed of light has been broken: Cutting-edge discovery in Plasma Physics<p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5VhUC0QDO7uRqpcgTGbKpu-jAB_GiTtTOkg4d9sEPWxW22gHYVnrxw5eWwxbTT5fdyn5fIC9hK_6KMn37Iz5xMcv7UQxDDN3JfLCX0g4m2KDmJwFKlwQbEFE4F7GppHujz9_N3OVt0QFc/s2048/lavi-perchik-vxn9hYL8lII-unsplash.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1365" data-original-width="2048" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5VhUC0QDO7uRqpcgTGbKpu-jAB_GiTtTOkg4d9sEPWxW22gHYVnrxw5eWwxbTT5fdyn5fIC9hK_6KMn37Iz5xMcv7UQxDDN3JfLCX0g4m2KDmJwFKlwQbEFE4F7GppHujz9_N3OVt0QFc/s320/lavi-perchik-vxn9hYL8lII-unsplash.jpg" width="320" /></a></span></div><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><br /><span style="color: #0b5394;"><br /></span></span><p></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b>Speed of light</b>, a
constant and we all know that. The constancy of the speed of light in a vacuum is a postulate
in the Special and General Theory of Relativity. Along with that, we can also calculate
the exact figure of the speed of light in a vacuum using laws of electromagnetism and
that figure is exactly <b>299,792 km/sec</b>.</span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">People always wondered
and asked that whether <b>is it possible to break the speed of light</b> and now we are
in the position to answer this question.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The breakthrough
discovery was done by the researchers from <b>Lawrence Livermore National
Laboratory in California, USA</b> and the <b>University of Rochester in New York, USA</b>.
<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Don’t get me wrong,
light has a constant speed limit but under certain conditions, this can be
broken for individual pulses of light. The researchers did the same thing. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">To achieve this feat,
they used <b>plasma</b>, the <b>fourth state of matter</b>. Plasma consists of a<b> gas of ions</b>.
Ions are basically particles, atoms or molecules with a net electric charge. A
common example of plasma is lightning. But it also can be artificially generated
by heating a neutral gas or subjecting it to a strong electromagnetic field. A
common example of its application is Neon light that we use for decoration and
lighting purposes.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">As we all know, <b>the speed
of pulses of light can be vastly different in different mediums and not the same as
the speed of light in a vacuum</b>. For example, the speed of light is slower in
water as compared to vacuum. The maximum speed of light which we can observe was the
speed of same in a vacuum and this has been broken in this experiment.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">In this experiment,
the scientists firstly created a hydrogen-helium plasma by ionizing a jet of the
gas with a polarized laser beam.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">I am not going too
deep into the theory polarization but for the sake of understanding this experiment,
I am giving an overall idea of it in simple terms. <b>Normally, when a light beam
travels, it has light waves at different planes and they have different angles
with each other. So, to get only one specific plane we use a method named as
Polarization, where, a device named a polarizer has a thin and closely
situated long chain of molecules to filer out a specific plane of light.</b></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"></span></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiT19i8cB0u3QdxtjVFBCoPPhr1xVyg3FPUHmoUs9GdVDnTV5u17WD2jY2sxGXlLAq4D4XU_bteuQyU_wPAUj_qFGK1kn4kC04sCNuVV2M5tLDcc_rMaF64PxcCgI4-uDjjxrSAhdGe01SJ/s772/772px-Vertical_polarization.svg.png" style="margin-left: auto; margin-right: auto;"><img alt="Speed of light has been broken: Cutting-edge discovery in Plasma Physics Example of Polarization. 1)Light Source 2)Unpolarized Light with all of its planes 3)Verticle Polarizer 4) Verticle Polarized light" border="0" data-original-height="600" data-original-width="772" height="249" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiT19i8cB0u3QdxtjVFBCoPPhr1xVyg3FPUHmoUs9GdVDnTV5u17WD2jY2sxGXlLAq4D4XU_bteuQyU_wPAUj_qFGK1kn4kC04sCNuVV2M5tLDcc_rMaF64PxcCgI4-uDjjxrSAhdGe01SJ/w320-h249/772px-Vertical_polarization.svg.png" title="Speed of light has been broken: Cutting-edge discovery in Plasma Physics Example of Polarization. 1)Light Source 2)Unpolarized Light with all of its planes 3)Verticle Polarizer 4) Verticle Polarized light" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Example of Polarization. 1)Light Source 2)Unpolarized Light with all of its planes 3)Verticle Polarizer 4)Verticle Polarized light</td></tr></tbody></table><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><br /><b><br /></b></span></span><p></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">After doing all these,
they aimed <b>a second laser beam</b> at the plasma and <b>where the two beams crossed</b>, a
peculiar thing happened, <b>the horizontal component of the second laser beam slowed
down as compared to the earlier state</b>. Now, as I said earlier that a specific beam
of light may have multiple planes at various angles to each other, so there is
also one horizontal plane or component and one vertical plane or component. So,
in this case, the horizontal component slowed down and it happened because of the
interaction between the two lasers and the plasma.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, when they measured
the <b>time delay between the horizontal and vertical components of the second laser
pulse, then they observed that they had a different speed and by tuning the
frequency (inverse of time) difference between the two beams, they found that
the speed can be tuned from 0.995c to between 0.12c and -0.34c and by this way
my friend, they found out that the pulse travelled faster than c. </b><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Fascinating right???<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">But after all these,
you might be thinking that how photons, that is, the quanta of light are
travelling faster than the speed of light?<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Well, it’s a classic
mistake. Notice carefully that at the beginning I warned you about this fallacy
and told you that its not the photon which is travelling faster than the speed
of light, rather its pulses. The velocity of pulses, which is technically
called group velocity, can vastly differ from the speed of light travels in
a vacuum. Photons are nothing but the effect of electromagnetic field according
to <b><a href="https://thedynamicfrequency.blogspot.com/search/label/Quantum%20Field%20Theory" target="_blank">Quantum Field Theory (QFT)</a></b> and<b> Quantum Electrodynamics.</b> So, it’s the electromagnetic
field wave for which the phenomenon of light happens. So, the speed of that
wave is c when it travels through a vacuum and the pulses of light made by it
can be changed in various mediums due to different group velocities. For example,
it slows down in medium like water. Just like that, the group velocity is
increased in the plasma medium.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><br /></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Applications</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">From an application
point of view, this experiment will significantly increase our knowledge in the
space of plasma physics. Moreover, the experiment will also lead to significant
<b>improvements in the field of laser technology</b>, which until now, mostly rely on solid-state
physics and optical materials for practical applications. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Using plasma to
amplify or change light characteristics will help us to create evermore
powerful lasers which can be used in applications like particle accelerators
and fusion tech.</span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">For those who want to
read the research paper on this experiment, click</span> <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.205001" target="_blank"><b>here</b></a>.<o:p></o:p></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing to our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article in mobile).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; 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Part 5 of the article series I wrote about Quantum Harmonic Oscillator. If you
have not read the <b><a href="https://thedynamicfrequency.blogspot.com/2020/10/quantum-harmonic-oscillator-intro.html" target="_blank">Part-1: Introduction to Quantum Harmonic Oscillator</a></b>, <b><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank">Part 2:Schrödinger’s Equation with Dimensionless Terms</a>,<a href="https://thedynamicfrequency.blogspot.com/2021/03/quantumharmonic-part3-asymptoticSol.html" target="_blank"> Part-3: The Asymptotic solution</a></b>
and <b><a href="https://thedynamicfrequency.blogspot.com/2021/04/quantumharmonic-part3-seriesSol.html" target="_blank">Part-4: The Series Solution of Schrödinger’s Equation</a></b>, then you cannot
understand what I am going to explain in this article, so reading those
articles is a must.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">In this article, I
will introduce you to <b>Hermite Polynomials</b>. Although, I will not discuss its
full details and normalization as its an advanced mathematical topic and out of
the scope of this article, rather you can just search it in the net there is
ample resources available about it. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Before moving forward
remember that, when <b>we introduced H in the previous article, we declared it as an
unknown variable</b>. In this article, we will try to find out more about that.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">So, if you find the
<b>family of solution of H</b></span></span><span style="font-family: Cambria, serif; font-size: 18px; text-align: center;"><span style="color: #0b5394;"><b>ₙ</b></span></span><span style="color: #0b5394;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>) to
equation 5 of my previous article, then those solutions are known as Hermite
Polynomials</b>. There are two major types of Hermite polynomial, one is
Probabilist’s Hermite Polynomial and another one is Physicist’s Hermite
Polynomial. Here we will use the second one. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">For example, I am
listing few of them below,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H</span><span style="font-family: "Footlight MT Light", serif; font-size: xx-small; line-height: 107%;">0</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) =1<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">H</span><span style="font-family: "Footlight MT Light",serif; font-size: xx-small; line-height: 107%; mso-bidi-font-family: Cambria;">1</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) = 2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H₂(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = 4</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> – 2<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H</span><span style="font-family: "Footlight MT Light", serif; font-size: 18px;">₃</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = 8</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background-color: white; font-family: Cambria, serif; font-size: 18px;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> – 12</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">One thing which you
can notice that the <b>highest power of </b></span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> appearing in the series for Hn(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) is </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξn</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">. If you want to get result in terms of real (x)
then just substitute </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">x</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, now, I hope you
got the idea about how to solve the equation 5 of previous article.<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, moving on to the
normalized harmonic-oscillator wavefunctions. It is given by<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ψ</span><span style="font-family: Cambria, serif; font-size: xx-small; line-height: 107%;">n</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = A</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="font-family: Cambria, serif; font-size: x-small;">n</span> H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background-color: white; font-family: Cambria, serif; font-size: 18px;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">)</span><span style="color: #0b5394;"> </span><span style="color: #0b5394;"> </span><b style="color: #0b5394;">(1)</b><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">where An is the normalization
factor,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">A</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: Cambria, serif; font-size: x-small;">n</span> = √[</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">α</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">/{(2^n) n! √π}]</span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">And where,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">
= (mk/</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ħ</span><span style="background-color: white; font-family: Cambria, serif; font-size: 18px;">²</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">)^1/4</span><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">Equation 1 here is
obtained from equation 1 of the series solution article.</span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMTmUtIeo-hpnJdSg0sfBudUUzYQifvVlNX4EcSw6WF7eMqcCd1xjgjXhHpbWLy65GkpOI3WYxU39cZsWiRPvSbAP8Jt9R67gSW92vUf5nOE-mU2dEbcYHIl9Ci_t45bQOPn7xuWKk-JQD/s1141/Harmonic_oscillator_potential_and_wavefunctions.png" style="margin-left: auto; margin-right: auto;"><img alt="Hermite Polynomial and Normalised Harmonic oscillator potential and wavefunctions Ratnadeep Das Choudhury the dynamic frequency" border="0" data-original-height="649" data-original-width="1141" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMTmUtIeo-hpnJdSg0sfBudUUzYQifvVlNX4EcSw6WF7eMqcCd1xjgjXhHpbWLy65GkpOI3WYxU39cZsWiRPvSbAP8Jt9R67gSW92vUf5nOE-mU2dEbcYHIl9Ci_t45bQOPn7xuWKk-JQD/w400-h228/Harmonic_oscillator_potential_and_wavefunctions.png" title="Hermite Polynomial and Normalised Harmonic oscillator potential and wavefunctions Ratnadeep Das Choudhury the dynamic frequency" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Harmonic oscillator wavefunctions for n=0,1,2,3</td></tr></tbody></table><br /><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; 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Part 4 of the article series I wrote about Quantum Harmonic Oscillator. If you
have not read the </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;"><a href="https://thedynamicfrequency.blogspot.com/2020/10/quantum-harmonic-oscillator-intro.html" target="_blank">Part-1: Introduction to Quantum Harmonic Oscillator</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;"> and </span><b style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;"><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank">Part2: Schrödinger’s Equation with Dimensionless Terms</a></b><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;"> and </span><a href="https://thedynamicfrequency.blogspot.com/2021/03/quantumharmonic-part3-asymptoticSol.html" style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;" target="_blank"><b>Part-3: The Asymptotic solution</b></a><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;">, then you cannot understand what I am going to explain in this
article, so reading those articles is a must.</span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Okay…without any
further delay let’s just start…<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The objective of this article is to find a general solution for the harmonic oscillator by finding
the series solution.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">From the <b>equation 7 of
my last article</b>, we obtained an expression, and to solve Schrodinger’s equation
for this problem, we wish to explicitly build in knowledge of the <b>exponential
asymptotic behaviour of </b></span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>ψ</b></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"> that was established in previous article.</span><o:p></o:p></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">So, there is a one way
to do so, that is by assuming that can be expressed as the <b>product of two
functions,</b> <b>one which will have the asymptotic behaviour of wavefunction and
other one is an unknown function, lets say it as H(</b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b>ξ</b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>)</b>. We can express what we said as follows,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ψ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">) <b>(1)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">This is just an
assumption for the solution taken by only<b> trial and error method</b>. We have to
see whether it satisfies our conditions or not.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">From elementary
calculus we know that any function of x ( here, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> can be expressed as an infinite polynomial using <b>Taylor or McLaurin
Series</b>, so, <o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = a</span><span style="font-size: 13.5pt; line-height: 107%;">₀
</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">+ a</span><span style="font-size: 13.5pt; line-height: 107%;">₁ </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + a</span><span style="font-size: 13.5pt; line-height: 107%;">₂ </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">² </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">+ … = </span><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: Cambria Math, serif; font-size: xx-small; line-height: 107%;">n</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> a</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-size: x-small;">n</span> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ </span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">[n ranges from 0 to ∞] <b>(2)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Here, <b>n ranges from 0 because we cannot
have negative powers for </b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b>ξ</b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b> as it would lead to infinities, which in this
case will lead to not normalizable</b>.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Now, we have to solve Schrodinger’s
equation for harmonic oscillator that is equation 11 of part-2 article this
this series of QHO.<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">d^2 </span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ψ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^2
+ ( </span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ </span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">– </span></b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^2)</span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ψ </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 0 <b>(3)</b></span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Differentiating
equation 1 we get, we get,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">d</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ψ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = [ d</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H/d </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> 2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> dH/d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + ( </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> 1)H ]</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">) <b>(4)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Note that we have
<b>dropped the explicit dependence of H on </b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b>ξ</b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b> for sake of clarity</b>.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, substituting
equation 4 and equation 1 in equation 3, we get,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">d</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H/d </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> 2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> dH/d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + (</span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> 1) H = 0<b> (5)</b></span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Now, we have
everything in our hands to attempt a series solution for H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">). So, by differentiating equation 2, we get,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">dH/d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = </span><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> na</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻¹</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> </span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">[n ranges from
0 to ∞]</span><span style="color: #0b5394;"> </span><span style="color: red;"><b> (6)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">again differentiating,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">d</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">H/d </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= </span><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1)a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻²</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"> [n ranges from 0 to ∞]</span><span style="color: #0b5394;">
</span><b><span style="color: red;">(7)</span></b><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Substituting equation 2, 6 and 7 in
equation 5, we get,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻² </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">+</span><span style="font-size: 13.5pt; line-height: 107%;">
∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> (</span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"> -</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">2n) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 0 </span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">[n ranges
from 0 to ∞] <b>(8)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Dealing with equation 8 will be hard as
expanding out the sums would give contributions of the form </span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">from both terms for i ≥ 2 and thus resulting into not all terms
having same power in </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b> collected together</b>. This
can be solved. Let’s see how…<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Firstly, see that the first sum will be
equal to 0 if n=0 or 1, so, we can write as follows,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻² </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 0 + 0 + </span><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻² <b> </b></span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>(9)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Where, <b>last term of RHS having n ranges
from n=2 to ∞ and in the term of LHS, n ranges from n=0 to ∞.</b><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">What we see in equation 9 is that,
here, <b>n is a dummy index</b>, which means that because the sum runs over all
possible integral values which are greater than 0 up to ∞, it is clearly
irrelevant what label is actually assigned to index of summation. <b>In short, we
can change or transform the index as we wish we solve the equation easily</b>. So,
let’s consider i = n </span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">–</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> 2,
to solve the problem we discussed two paragraphs above. Now equation 9 converts
into,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻² </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= </span><span style="font-size: 13.5pt; line-height: 107%;">∑ᵢ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> (i+2) (i+1)
a</span><span style="font-family: Cambria Math, serif; font-size: xx-small; line-height: 107%;">i+2</span></span><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"> <b>(10)</b></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Where,<b> in the term of LHS, n ranges
from n=0 to ∞ and in the term of RHS, i ranges from i=0 to ∞</b><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Now, in equation 10 both n and i are
dummy indexes, so, we can define n=i,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ⁻² </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= </span><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> (n+2) (n+1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n+2</span> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"> <b>(11)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">If you have a doubt on equation 11,
then just expand the equation on both sides, you will find they are absolutely
identical.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">So, now we have solved our problem
which I discussed a few paragraphs above and thus now from equation 8 and 11,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt; line-height: 107%;">∑</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span> {(n+2) (n+1) a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n+2</span> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">+</span><span style="font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"> -</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">2n) a</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span>}</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 0 </span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">[n ranges from 0 to ∞]
<b>(12)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">From, equation 12 we can see that the
terms inside the curly braces are pure numbers and don’t have any dependence on
</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">. Also, the whole sum results in 0, and from our knowledge we
know that in general, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">is not equal to 0, but
still, the sum is 0. The only way <b>this can happen if every term inside curly
braces vanishes</b>, so this can be achieved by,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n+2</span> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= {(2n+1</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)/(n+2) (n+1)}</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> a</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-size: x-small;">n+2</span> </span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>(13)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Now, from our knowledge from series
solution theory, we know that<b> equation 13 is called as a recursion relation</b> as
it specifies a given <b>expansion coefficient recursively in terms of the
preceding coefficient in the series</b>. So, we need to find </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">a</span><span style="font-size: 13.5pt; line-height: 107%;">₀</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> and a</span><span style="font-size: 13.5pt; line-height: 107%;">₁</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> terms only and then we can find the even and odd terms on the
basis of </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">a</span><span style="font-size: 13.5pt; line-height: 107%;">₀ </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">and a</span><span style="font-size: 13.5pt; line-height: 107%;">₁ </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">respectively. <o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Before we progress further, I want to
remind you guys that <b>we have a boundary condition that is the wavefunction must
be finite as </b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b>ξ </b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b>tends to ±∞</b>. So, in the equation
1, we know that </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> is convergent as we derived that in this way in our part-3
article on asymptotic solution. Thus, a<b>s long as </b></span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) is convergent or at worst case, if it diverges
no more strongly than the exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">+</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>), then the overall solution will remain convergent</b>.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">But, unfortunately, if
you consider a general series expansion of then you will find it behaves
asymptotically like exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">). General series
expansion of exp(</span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">),<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">exp(</span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = 1 + </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²ⁱ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2! + </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="font-size: 13.5pt; line-height: 107%;">³</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">³ⁱ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">/3! + …
<b>(14)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, from equation 2
and 13,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Term(n+1) / Term(n) = (a</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n+2</span> </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/ a</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Cambria Math", serif; font-size: x-small;">n</span>) </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">{(2n+1</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">)/(n+2) (n+1)} <b>(15)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">And the ratio of successive terms in
the exponential function of equation 14,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Term(n+1) / Term(n) = </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/(</span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">+1) <b>(16)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Equation 15 and 16 at
n tends to </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">∞,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Term(n+1) / Term(n) ~ 2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">/n <b> (17)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">And, <o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Term(n+1) / Term(n) ~ </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">/n <b>(18)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">On comparing equation
17 and 18, we get, </span><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">=2 and i=2. Thus, we conclude that harmonic
oscillator solution behaves as an exponential of the form <b>exp(</b></span><b><span face="Arial, sans-serif" style="background: white; font-size: 13.5pt; line-height: 107%;">η</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁱ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) =
exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>)</b>. This means that we can represent wavefunction
as,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ψ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) = H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) ~ e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">2</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">) ~ e^(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">3</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">)
<b>(19)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Equation 19 is clearly<b>
divergent and non-normalizable</b>. The only way to solve this problem and prevent
this catastrophe is to prevent the series for H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) from going to an infinite number of terms. Thus, terminating the
series of H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) after a finite number of terms.<o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">So, if the highest
power of </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> appearing in H(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) is n, and if all a</span><span style="font-size: 13.5pt; line-height: 107%;">ᵢ </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 0 for i>n, then from
the recursion relation (equation 13),<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">λ</span></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"> = 2n+1 </span><span style="color: #0b5394;"> <o:p></o:p></span></span></p><p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">So, from above equation and equation 2
and 10 from Part-2 article that is </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Schrödinger’s Equation with Dimensionless Terms, we can find the
energy corresponding to quantum number n:<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">λ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = 2n+1 = </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">2E / </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"> </span>
</span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> <o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">or,<o:p></o:p></span></span></p><p align="center" class="MsoNormal" style="text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">E = (n+1/2)</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ħ</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = (n+1/2)</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(√k/m), n=1,2,3,4,… <b> (20)</b></span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">From equation 20, <b>we can see that E is
proportional to n, which indicates that the harmonic potential energy levels
are equally spaced</b>. These equally spaced energy levels are known as<b> vibrational
levels</b> as the harmonic potential mimic the attractive forces between the
molecules. <o:p></o:p></span></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Apart from these, if
you observe a spectrum then you will find that molecular spectra exhibit lines
between the equally spaced vibrational transitions. These are called <b>rotational levels</b>. I will someday write a brief article on this topic.<o:p></o:p></span></span></p><p>
</p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, that’s it, if you
survived till the last then congrats as not as many people can do so. In my
next article, which will be<a href="https://thedynamicfrequency.blogspot.com/2021/05/quantum-oscillator-part5-hermitepoly-normalised.html" target="_blank"> <b>part-5 in this series, I will write on </b></a><span style="background: white;"><b><a href="https://thedynamicfrequency.blogspot.com/2021/05/quantum-oscillator-part5-hermitepoly-normalised.html" target="_blank">Hermite Polynomials and Quantum Harmonic
Oscillator Wavefunctions</a></b>. So, stay tuned for that. Once it is uploaded, the
link will be available here.</span><o:p></o:p></span></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing to our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar or in the bottom menu bar if you are reading this article in mobile).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; 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Check it now on <b><a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank">YouTube</a></b></i></span></div></i></span></div></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div><div><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"><br /></span></div></div></div><p><br /></p>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-43584563645285279702021-03-18T23:13:00.005+05:302021-05-09T01:07:19.365+05:30Quantum Harmonic Oscillator Part-3: The Asymptotic Solution of Schrödinger’s Equation Quantum Harmonic Oscillator Potential<p></p><p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><div class="separator" style="clear: both; text-align: center;"><br /></div><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDOq1G0luugKBkSfqeCXvw1zCFk2kpafpv7mkXMCCNvkAuW_wjy5_3nJOTzHU0N-nZCVaeVvt_ZrsFXsvMROtHIkp30A-z0W1oZd78RuYI73p3C4YpHAZnd8YmwTtw5WwjecLP5q7myfZU/s449/449px-Excited_state_for_quantum_harmonic_oscillator.svg.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="278" data-original-width="449" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDOq1G0luugKBkSfqeCXvw1zCFk2kpafpv7mkXMCCNvkAuW_wjy5_3nJOTzHU0N-nZCVaeVvt_ZrsFXsvMROtHIkp30A-z0W1oZd78RuYI73p3C4YpHAZnd8YmwTtw5WwjecLP5q7myfZU/s320/449px-Excited_state_for_quantum_harmonic_oscillator.svg.png" width="320" /></a></div><br /><span style="color: #0b5394;"><br /></span></span><p></p><p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">This article is Part 3 of the article series I wrote about Quantum
Harmonic Oscillator. If you have not read the <b><a href="https://thedynamicfrequency.blogspot.com/2020/10/quantum-harmonic-oscillator-intro.html" target="_blank">Part-1: Introduction to Quantum Harmonic Oscillator</a></b> and <b><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank">Part 2: Schrödinger’s Equation with Dimensionless Terms</a></b>, then you cannot understand what I am going to explain in this article,
so reading those articles is a must. Also, this article is a bit technical
and a lot more mathematical so, having knowledge of calculus and solution of
equations is a necessary skill to proceed.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Okay, then… let’s start and tame the beast…<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">In my previous article on Schrödinger’s Equation with Dimensionless
Terms, we concluded with a beautiful equation which is Schrodinger’s
equation with two dimensionless variables (see <b>equation 11 in my part 2
article</b>).<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">We are going to use that equation here.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Our task is to solve that equation for </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) and then
revert the solution to x-space by the substitution, <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="mso-tab-count: 1;"> </span></span><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">x<span style="mso-spacerun: yes;"> </span><b>(1)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">We are going to use the asymptotic
method to find a solution in this article. In the next article, <b>we will discuss a
more detailed general series solution</b> for this problem. After that, we will
discuss the <b>characteristics of the wave function in my article on Hermite
Polynomials and Quantum Harmonic Oscillator Wavefunctions</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">If you look closely at the equation (11)
of the previous article then you will find that </span><b><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)
approaches zero as </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b> or x
(from equation 1) tends to ±∞</b>. This lead to the formation of the asymptotic form of
</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">. By
this way we can find an expression that gives us the solution for large values
of </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ, that is <b>ξ</b></span><b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">--</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>±∞</b><span style="background: white;">.</span>
The problem then<b> reduces for finding a solution for small values of </b></span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> to be grafted onto the asymptotic solution to
establish a solution that will be valid for all values of </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">. That will be
done in my next article on the series solution.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">For now, to establish an asymptotic solution
for </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">, take equation 11 of the last article when </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> is large. By doing
so we will find that only </span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">² </span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b>is large and will dominate</b>,
so that </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Schrödinger</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> equation
will become like this,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">d</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>~ </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span></span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">(2)</span><span style="background: white; color: #0b5394;"><o:p></o:p></span></span></b></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, we can postulate a trial solution for the above equation,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span><span style="color: red;"><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) ~ A exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ</span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">)<span style="mso-spacerun: yes;"> </span></span><b><span style="color: red;">(3)</span><span style="color: #0b5394;"><o:p></o:p></span></b></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">where, A, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> and n are unknowns,
and we have to determine their values.</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, now
differentiating the trial solution, we get,<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p><span style="color: #0b5394;"> </span></o:p></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">d</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = [n</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">ⁿ⁻</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² + n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1)</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> β</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">ⁿ⁻</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²]</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">)<span style="mso-spacerun: yes;"> </span><b>(4)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 353.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, after substituting this solution, equation 4, back to the equation
2, we get,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 353.1pt; text-align: center;"><span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394; mso-spacerun: yes;"> </span><span style="color: red;">n</span></span><span style="color: red;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">ⁿ⁻</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² + n(n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1)</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> β</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">ⁿ⁻</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² ~ </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;"><span style="mso-spacerun: yes;"> </span><b>(5)</b></span><span style="mso-tab-count: 1;"><span style="color: red;"> </span><span style="color: #0b5394;"> </span></span><span style="color: #0b5394;">`</span><span style="background: white; color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">From, equation 3, which is our trial solution we can
see that <b>n must have a positive value as </b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b>ξ</b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b> tends to ±∞</b>. So, for a large value of the first term of the <b>left-hand side
will dominate</b>, and in this way we get,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="text-align: center;"><span><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">n</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Times New Roman", serif; font-size: 13.5pt; line-height: 107%;">ⁿ⁻</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² ~ </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²<span style="mso-spacerun: yes;"> </span></span><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(6)</span></b></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">By comparing, 2n</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">2 = 2, we get <b>n=2</b>. Also by comparing, </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">n</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² = 1, and since we got n=2, so, 4</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> β</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">² = 1, and
thus we get </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= ±1/2. Before we
pick up the value for </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">, we have to know what is meant by normalizing. <b>Normalizing a
wave function means that we have to find an exact form which ensures that the
probability that the particle is found somewhere in space is 1</b>. This shows us
that, the particle must be somewhere. This means in technical terms as to solving
for some constant subject to the above constraint that the probability is equal
to 1. <o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Now, as you know the importance of normalizing, so on the basis
of that we can choose the value of </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">. <b>Choosing </b></span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> =+1/2 will be catastrophic
as it would lead to the asymptotic solution of the </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) ~ A exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">ⁿ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>), which
is impossible to normalize</b>. Rather if we choose </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">β</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> =</span><span style="font-family: "Cambria Math",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">-</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">1/2 then, <b>we can normalize it as something to the
value of negative number will always finite and will not tend to infinity, so
there is always a chance that we get a value</b>. Try it yourself if you don’t
agree!!!<o:p></o:p></span></span></p>
<p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, in this way we finally
found our Asymptotic solution for SE of QHO potential,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span><span style="color: red;"><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) ~ A exp(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">-</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="font-size: 13.5pt; line-height: 107%;">²</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/2</span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">)<span style="mso-spacerun: yes;"> </span></span><b><span style="color: red;">(7)</span><span style="color: #0b5394;"><o:p></o:p></span></b></span></span></p>
<p class="MsoNormal"><span><span style="color: #0b5394; font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">And remember </span><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">--</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">±∞</span><span style="background: white; color: #0b5394;">.<o:p></o:p></span></span></span></p>
<p class="MsoNormal"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Equation 7 is the only possible solution to equation 3 which
can satisfy equation 2 </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">or in
other words reduced dimensionless SE equation for QHO potential, <span style="background: white;">when </span></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">--</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">±∞. This is only a partial solution as I wrote
earlier, we still have a task left to formulate a generalized equation for the wavefunction. This will be done in the next article.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">My next article will be on the <b><a href="https://thedynamicfrequency.blogspot.com/2021/04/quantumharmonic-part3-seriesSol.html" target="_blank">series solution for harmonic potential (part-4 of this article series)</a></b> which will be followed by an article on <a href="https://thedynamicfrequency.blogspot.com/2021/05/quantum-oscillator-part5-hermitepoly-normalised.html" target="_blank"><b><span style="background: white;">Hermite Polynomials and Quantum Harmonic
Oscillator Wavefunctions </span>(part-5 of this article series)</b></a><span style="background: white;">. </span>Once its uploaded, the link
will be uploaded here.<o:p></o:p></span></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing to our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; 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Harmonic Oscillator. If you haven’t read the <b><a href="https://thedynamicfrequency.blogspot.com/2020/10/quantum-harmonic-oscillator-intro.html" target="_blank">part-1 of it where I introduced the topic</a></b>, then it will be for you to challenging understand this article.
So, I highly suggest you read that article first and then
proceed in this article.</span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, let’s get started with this article….<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Almost any incarnation of Schrödinger’s equation can be made dimensionless
by finding two combinations. The first combination includes the mass of the particle
(m), </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(reduced Planck’s
constant which is h/2π) and a constant with a dimension of reciprocal length (let’s
say it as </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">. Another
combination includes the mass of the particle (m), </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">and a constant with a dimension of reciprocal
energy (let’s say it as </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)</span></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">. Then we define dimensionless variable,</span><span style="color: #202122;"><o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">x<span style="mso-spacerun: yes;"> </span><b>(1)</b><o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">λ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> <span style="background: white;">=</span> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">E <b>(2)</b></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x and E are in Schrödinger’s Equation are the cast items of </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> and </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">λ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The result is a dimensionless differential equation that is free of
constants, consequently easier to manipulate than the original version. This has
been done because, in general, dimensionless formulations are amenable to computation
as we don’t have to deal with units and quantities which are exceptionally large
or small in magnitude. Once we get the solution of the dimensionless
differential equation, they can be put back in terms of original variables by
simple substitutions.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">As we have discussed in my Part-1 that harmonic oscillator potential has
only one constant k so we begin by finding a combination of m, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">and k that has units of reciprocal meters or
inverse meters (finding for </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">). We will use simple dimension analysis to find the powers a,
b and c for m (unit - kg), k (unit -kg/sec^2) and </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"> (unit - joule/sec
or kg meter^2/sec) respectively.</span><span style="color: #202124;"><o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> (m^a) (k^b) (</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^c)<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">meter^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1 </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> [(kg)^a] [(kg / sec^2)^b] [(kg meter^2
/ sec)^c]<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">meter^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1 </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> [kg^(a+b+c)] [meter^2c] [sec^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">(2b +c)]<span style="mso-spacerun: yes;"> </span></span><b><span style="color: red;">(3)</span><span style="background: white; color: #202124;"><o:p></o:p></span></b></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">so, when we compare LHS and RHS of equation 3, we get,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">2c = <b>–</b>1<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">So, c = <b>–</b>1/2<span style="mso-spacerun: yes;"> </span><b>(4)</b></span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Also,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">2b + c = 0<span style="mso-spacerun: yes;">
</span><b>(5)<o:p></o:p></b></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">and, a + b + c = 0<span style="mso-spacerun: yes;"> </span></span><b><span style="color: red;">(6)</span><o:p></o:p></b></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, after solving equation 5 and 6 and putting the value of c from
equation 4, we get,</span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p> </o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">a = 1/4 <o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">b = 1/4</span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α </span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">=</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">(m^a) (k^b) (</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^c) = (m^1/4) (k^1/4) (</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> ^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1/2) <o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = (m k / </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> ^2)^1/4 <b>(7)</b></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">similarly, we will also find for reciprocal energy or inverse joules,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> (m^a) (k^b) (</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> ^c)<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">joule ^<b>–</b>1 </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> <span style="background: white;">[(kg)^a] [(kg / sec^2)^b]
[(kg meter^2 / sec)^c]<o:p></o:p></span></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">[(sec^2)/(kg
meter^2)] </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">≡ </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">[kg^(a+b+c)] [meter^2c] [sec^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">(2b +c)]<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><b>(8)</b></span><span style="background: white; color: #202124;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">So, by comparing LHS and RHS of equation
8, we get,</span><span style="color: #202124;"><o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">a = 1/2<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">b = –1/2<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">c = –1</span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">=</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> (m^a) (k^b) (</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> ^c) = <span style="background: white;">(m^1/2) (k^</span><b>–</b><span style="background: white;">1/2) (</span></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> ^</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">–</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">1)<span style="mso-spacerun: yes;">
</span><o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> = (1 / </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)√(m / k)</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> <o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span> <span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 1 / </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> <b>(9)</b></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">where, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span> <span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">= <span style="background: white;">√ (k / m) is the angular frequency of a
classical mass/spring system.<o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">For algebraic convenience, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> can be defined with
an additional dimensionless factor of 2.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, equation 9 changes to,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ε</span> <span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= 2 / </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ħ</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> <b>(10)</b></span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">After this we define dimensionless variables, as we did in equation 1
and 2.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Then, substituting them into Schrödinger’s equation for the harmonic
oscillator, and remember to make the substitution in the derivative as well, after
which I get,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">d^2 / dx^2 = </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">α</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^2 (d^2 / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;">^2)</span><span style="color: #202122;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Thus, reducing </span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Schrödinger’s equation,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 194.1pt; text-align: center;"><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">d^2 </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">ψ</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> / d</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"> ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^2<span style="mso-spacerun: yes;">
</span>+ ( </span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">λ</span> <b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">– </span></b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ξ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">^2)</span><span style="font-family: "Cambria",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> ψ</span>
</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;">= 0<span style="mso-spacerun: yes;"> </span></span><b><span style="color: red;">(11)</span><o:p></o:p></b></span></p>
<p class="MsoNormal" style="tab-stops: 194.1pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">This is a very cleaner equation as compared to the original one. We
will use equation 11 in our <b>Part 3 of QHO</b>. In <b>part 3</b>, we will dig into the
<b><a href="https://thedynamicfrequency.blogspot.com/2021/03/quantumharmonic-part3-asymptoticSol.html" target="_blank">concept of Asymptotic Solution of QHO</a></b>. Once it's uploaded, the link will be
uploaded here.</span><o:p></o:p></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; 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<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">In this article, I going to explain the <b>invariancy of Spacetime
interval equations in a mathematical way</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">But before you continue, you have to read my articles on <a href="https://thedynamicfrequency.blogspot.com/2020/12/a-crisped-introduction-of-spacetime.html" target="_blank"><b>Introduction to Space-Time</b></a> and <b><a href="https://thedynamicfrequency.blogspot.com/2021/01/lorentz-transformation-simplified.html" target="_blank">Lorentz Transformation</a></b> to understand this article in a better
way.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Let’s get started….<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">As I have explained <b>in my article on Introduction to Space-time</b>, as it
sounds, made up of two critical words, <b>space and time</b>. <b>Space</b>, as per the
<b>traditional physics</b>, represents the<b> 3-dimensional coordinate system of our
universe</b>. Scientists <b>before Einstein</b> believed that our universe is <b>limited to
these 3 coordinates (x, y and z-axis)</b> and the universe according to this
convention is called <b>Newtonian</b>. But, in <b>1905</b>, Einstein showed in his<b> Special
Theory of relativity</b> that <b>time is also an integral factor</b>. Well, it was not
before <b>1908</b>, when one of the Einstein’s own professor and renowned physicist
<b>Herman Minkowski</b> proposed the concept of the <b>union of space and time</b> as a way to
<b>reformulate Special Relativity and by this way, the concept of space-time was
born</b>.<o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqx6i5vzBGonev6yIQPmFydr5F0SNPqEgtLIicb9juYCcAJnJaVtTgBT1IYaHl7esJHqvEfsCFUVx4D4Wduwn5pLseP0X_YnZOkMJoHaD2uRiBIZn9bvFu2zg8awiDfA09YHz5Vgnp87jW/s1152/lorentz+pic.png" style="margin-left: 1em; margin-right: 1em;"><img alt="Understanding the Invariancy of Space-Time Interval Equations with Mathematical Proof The dynamic frequency ratnadeep das choudhury" border="0" data-original-height="648" data-original-width="1152" height="221" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqx6i5vzBGonev6yIQPmFydr5F0SNPqEgtLIicb9juYCcAJnJaVtTgBT1IYaHl7esJHqvEfsCFUVx4D4Wduwn5pLseP0X_YnZOkMJoHaD2uRiBIZn9bvFu2zg8awiDfA09YHz5Vgnp87jW/w393-h221/lorentz+pic.png" title="Understanding the Invariancy of Space-Time Interval Equations with Mathematical Proof The dynamic frequency ratnadeep das choudhury" width="393" /></a></div><p></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">To derive the equations to prove its invariancy, we need to understand
the above picture. Here in this picture, there are two coordinate systems, <b>S
and S’</b>. <b>Space and time coordinate of S are x, y, z and t, </b>and for<b> S’ they
are x’, y’, z’ and t’</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, suppose, in the course of motion of S’ relative to S in the
x-direction with a <b>speed v</b>, a flash of light is emitted at the origins of S and
S’ when they were just coincident. Let’s identify <b>that instant as t=0=t’</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Using this picture, <b>I derived the Lorentz transformation equations in
my Introduction to Lorentz Transformation article</b> and in this article, we are going
to f<b>ind the space-time interval equation using it</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;"><b>Proof of Invariancy</b></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">As we all know,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Time = Distance / Velocity<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t = √ (x</span><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 107%; mso-bidi-font-family: Cambria;">²</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) / c<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">c</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> = x</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b> (1)<span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Distance = √ (x<span style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></span> + y<span style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></span> + z<span style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></span>), because it has 3 axes.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, to prove invariancy, we need to show,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b>x</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + y</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + z</b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> – c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> t</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> = x’</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + y’</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b>+ z’</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> – c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> t’</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> (2)</b><span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Why??? Because invariancy means in every condition, the space-time
interval equation in both S and S’ will be equal. Which means that we can write
the space-time interval equation for S and S’ plane and equate them as we did
in equation 2.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, proving LHS = RHS,<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Where, </span>LHS = <b>x</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + y</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + z</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b> <b>– c</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> t</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> </b><span style="color: #0b5394;"> </span><span style="color: #0b5394;">(space-time interval equation in S plane)<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">And, </span>RHS = <b>x’</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + y’</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> + z’</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b> <b>– c</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> t’</b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="color: #0b5394;"> </span></b><span style="color: #0b5394;"><o:p></o:p></span></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 18px;">(space-time interval equation in S' plane)</span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, using Lorentz Transformations in RHS,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p><span style="color: #0b5394;"> </span></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">[x – vt / (√ 1 – v</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">/c</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">)]</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">² </span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">+ y</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + z</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b> <b>– c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> [(t – vx/c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b>) / (√ 1 – v</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b>/c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b>)]</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="mso-spacerun: yes;"> </span></b>(3)<span style="color: #0b5394;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Since, <o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x’ = x – vt / (√ 1 – v</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">y’ = y<span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">z’ = z<span style="mso-spacerun: yes;">
</span><o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t’ = (t – vx/c</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) / (√ 1 – v</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<span style="color: #0b5394;"><o:p></o:p></span></span></b></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now if you solve equation 3, you will get,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">x</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + y</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> + z</span></b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b> <b>– c</b><b style="font-family: "Times New Roman"; font-size: medium;"><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b><b> t</b></span><b><span style="font-family: "Footlight MT Light",serif; font-size: large; line-height: 25.68px; mso-bidi-font-family: Cambria;">²</span></b></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">which is our LHS.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Thus, </span>LHS = RHS<span style="color: #0b5394;"> and<b> invariancy of the space-time interval is proved.</b><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>Same can also be proved by inverse Lorentz Transforms and putting
those in LHS and then deriving RHS from it. The choice is yours always!!!</b><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, that’s its guys…that’s all about Space-time intervals and its
invariancy.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">In my upcoming articles, I will write about</span><b><span style="background: white; border: 1pt none windowtext; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%; padding: 0cm;"> 4-vectors, tensors</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> etc. are very important to understand
relativity and cosmology in general. So, I will be writing about these gems,
very soon…</span></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px;"><span style="-webkit-text-stroke-width: 0px; float: none;">Once these articles will be available links will be provided here...</span></span></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and <b>follow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar if you are viewing in desktop and if you are viewing from mobile then its in the bottom of the page).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><br /></div><div class="MsoNormal" style="text-align: justify;"><span style="font-family: "footlight mt light" , serif; 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font-size: 13.5pt;"></span></span></div><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><div style="font-family: "Times New Roman"; font-style: normal; margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>I also have a youtube channel where I make videos on Productivity & Growth, Physics and Technology. Check it now on <b><a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank">YouTube</a></b></i></span></div><div><br /></div></i></span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-51888397156108115042021-01-24T23:49:00.010+05:302021-02-07T13:13:50.054+05:30Lorentz Transformation: A Simplified Overview<p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHdz_54cW6aYTvkZtET-aT15ZENAFUzO8yFk3HjJjftwFCYixXlMWIFBzEXAjcb5ujZWLzVApW1CGhR2lsxnURtZ963s2avryBFPQA4bwoV-QKdboXotCmh-On3oBi3UGyCC8yVcdyxYPl/s300/Animated_Lorentz_Transformation.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHdz_54cW6aYTvkZtET-aT15ZENAFUzO8yFk3HjJjftwFCYixXlMWIFBzEXAjcb5ujZWLzVApW1CGhR2lsxnURtZ963s2avryBFPQA4bwoV-QKdboXotCmh-On3oBi3UGyCC8yVcdyxYPl/s0/Animated_Lorentz_Transformation.gif" /></a></div><br /><span style="color: #0b5394;"><br /></span><p></p><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Okay…. there are already a lot of articles and books have been written
on this topic, but still, I need to cover this for two main reasons. First, my upcoming
articles will be based on this topic and I don’t want my readers to have
trouble while finding adequate material to understand this topic. Second, as I
have also studied many articles and books about this topic, so while reading
them I always felt that there is always the ‘student’s side’ of the explanation
was missing from the text. So, in this article I will try to include that.</span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Lorentz transform plays a critical and fundamental role in the space
of relativity. Without it you cannot derive the expressions for the popular
theories like length contraction, time-dilation and popular equation of mass-energy
relation, e = mc^2. So, by this I hope you can understand the gravity this
topic holds.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The Lorentz transformations are named after the Dutch physicist <b>Hendrick
Lorentz</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Before, we start the derivation of the transformation, <b>we need to keep
two postulates in our memory</b>: <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;"><b>1)</b></span><span style="color: #0b5394;"> These transformations will follow <b>the principle of relativity</b>. The principle
of relativity states that, all laws of classical mechanics are valid in any two
coordinate systems moving uniformly relative to each other<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b><span style="color: red;">2)</span></b><span style="color: #0b5394;"> The universal <b>constancy of the speed of light</b> in vacuum will not be
violated<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, now you are ready to jump into the juicy derivation part….<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Derivation of Lorentz transformations</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Suppose, there are two co-ordinate systems, S and S’. The space and
time coordinate of S are x, y, z and t, and for S’ they are x’, y’, z’ and t’.<o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLy5p8s7r1YRS84-hQp6b9BELhnb-htFRLMF4xqUUiF_bzDiwO5XAg1CllNsiPf1HYPbJq-ql6q76CQ3F3Mk7ux0xMlekWTSvCWy1KR7gIWurogJf3-LuxHajlYahE9vEmPSCb08Bbi27u/s1152/lorentz+pic.png" style="margin-left: 1em; margin-right: 1em;"><img alt="Lorentz Transformation: A Simplified Overview The Dynamic Frequency Ratnadeep Das Choudhury" border="0" data-original-height="648" data-original-width="1152" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLy5p8s7r1YRS84-hQp6b9BELhnb-htFRLMF4xqUUiF_bzDiwO5XAg1CllNsiPf1HYPbJq-ql6q76CQ3F3Mk7ux0xMlekWTSvCWy1KR7gIWurogJf3-LuxHajlYahE9vEmPSCb08Bbi27u/w400-h225/lorentz+pic.png" title="Lorentz Transformation: A Simplified Overview The Dynamic Frequency Ratnadeep Das Choudhury" width="400" /></a></div><span style="color: #0b5394;"><br /></span><p></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, suppose, in the course of motion of S’ relative to S in the
x-direction with a speed v, a flash of light is emitted at the origins of S and
S’ when they were just coincident. Let’s identify that instant as <b>t=0=t’</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, for each observer, the flash of light would <b>spread out like a
spherical wave with speed c according to two postulates,</b> so for each wave,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">Time = Distance / speed<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t = √ (x</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) / c<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> = x</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="mso-spacerun: yes;"><span style="font-family: "Times New Roman"; font-size: x-large;">²</span> </span><span style="mso-spacerun: yes;"> </span>(1)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Distance = √ (x<span style="font-family: "Times New Roman"; font-size: x-large; text-align: center;">²</span> + y<span style="font-family: "Times New Roman"; font-size: x-large; text-align: center;">²</span> + z<span style="font-family: "Times New Roman"; font-size: x-large; text-align: center;">²</span>), because it has 3 axes.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>Equation 1 is called a Spacetime interval equation</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, as the spacetime interval is invariant, we can write,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z</span></b><span style="font-size: x-large;">²</span><b><span face=""Calibri Light",sans-serif" style="font-size: 13.5pt; line-height: 107%; mso-ascii-theme-font: major-latin; mso-bidi-theme-font: major-latin; mso-hansi-theme-font: major-latin;">-</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> =
x’</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + y’</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> + z’</span></b><span style="font-size: x-large;">²</span><b><span face=""Calibri Light",sans-serif" style="font-size: 13.5pt; line-height: 107%; mso-ascii-theme-font: major-latin; mso-bidi-theme-font: major-latin; mso-hansi-theme-font: major-latin;">–</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t’</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> = 0</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="mso-spacerun: yes;"> </span>(2)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>I will write another article to prove equation 2, so, stay tuned for
that…</b><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Since the relative motion occurs in the x-direction, the y and z
coordinates should remain unchanged,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">y = y’<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">z = z’<o:p></o:p></span></b></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, after substituting this in equation 2, we get,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span></b><b><span face=""Calibri Light",sans-serif" style="font-size: 13.5pt; line-height: 107%; mso-ascii-theme-font: major-latin; mso-bidi-theme-font: major-latin; mso-hansi-theme-font: major-latin;">-</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> = x’</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span></b><b><span face=""Calibri Light",sans-serif" style="font-size: 13.5pt; line-height: 107%; mso-ascii-theme-font: major-latin; mso-bidi-theme-font: major-latin; mso-hansi-theme-font: major-latin;">–</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t’</span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="mso-spacerun: yes;"><span style="font-family: "Times New Roman"; font-size: x-large;">²</span> </span>(3)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, from the figure, at O’, x’ should be 0 but x is vt and at O, x is
0 but x’ is -vt. So, we can write,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x’ = k (x – vt)<span style="mso-spacerun: yes;">
</span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(4)<b><o:p></o:p></b></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x = k’ (x’ – vt’) <span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(5)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">where, k and k’ are the constants and added for <b>precise measurement</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">From eq. (4) and (5), we eliminate x’,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t’ = k [ t – (x/v)(1–1/kk’)] <span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(6)<b><o:p></o:p></b></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, from eq. (3), (4) and (6),</span><o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span face=""Calibri Light",sans-serif" style="font-size: 13.5pt; line-height: 107%; mso-ascii-theme-font: major-latin; mso-bidi-theme-font: major-latin; mso-hansi-theme-font: major-latin;">-</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span></b><b><span style="background: white; color: #202124; font-family: "Times New Roman",serif; font-size: 10.5pt; line-height: 107%;">≡</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> (k</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)(x</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> – 2xvt + v</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> t</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) – c</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> k</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> [t</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> – (2xt/v)(1 – 1/kk’) + (x</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/v</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)(1 – 1/kk’)</span></b><span style="font-size: x-large; text-align: center;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">]<span style="mso-spacerun: yes;"> </span>= 0<span style="mso-spacerun: yes;">
</span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(7)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">By equation the coefficients of t^2, as eq. (7) is an identity and true
values for all values of x and t, we get,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">– c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> = k</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> – k</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> </span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">k = 1 / √ (1 </span></b><b><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px;">–</span></b><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> / c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> )<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(8)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">and this equation is known as <b>Lorentz Factor</b></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">and by equating coefficient of xt,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">0 = –2k</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">v + 2(c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> k</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> / v) (1 – 1/k k’)<span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(9)<o:p></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now, using Eq. 8 and 9 in Eq.<span style="mso-spacerun: yes;"> </span>4
and 6, we get ,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x’ = x – vt / (√ 1 – v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(10)<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">y’ = y<span style="mso-spacerun: yes;">
</span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(11)<o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">z’ = z<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(12)<b><o:p></o:p></b></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t’ = (t – vx/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) / (√ 1 – v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">(13)<b><o:p></o:p></b></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Eq. 10, 11, 12 and 13 are known as<b> Lorentz transformation or Einstein –
Lorentz Transformation</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Also, we can obtain the r<b>everse transformation</b> if we replace x’ with x
and t’ with t,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x = x’ – vt’ / (√ 1 – v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<span style="mso-spacerun: yes;"> </span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">y = y’<span style="mso-spacerun: yes;">
</span></span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">z = z’<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t = (t’ – vx’/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">) / (√ 1 – v</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">/c</span></b><span style="font-size: x-large;">²</span><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">)<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></b></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Now from Lorentz and its reverse transforms, if <b>v << c </b>then<b> we can ignore the fraction involving v and c as its value will be much lower than one and subtracting it from 1 will give a result which will be approximately equal to 1 </b>and thus we
can also obtain the following equations which are also known as <b>Galilean transformation</b>,</span><o:p></o:p></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">x’ = x – vt<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">y’ = y<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">z’ = z<o:p></o:p></span></b></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">t’ = t<o:p></o:p></span></b></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">It’s interesting, isn’t ???<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">In my upcoming articles, I will write about </span><b><span style="background: white; border: 1pt none windowtext; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%; padding: 0cm;"><a href="https://thedynamicfrequency.blogspot.com/2021/01/spacetime-interval-invariant.html" target="_blank">spacetime interval and how its invariant</a>,
4-vectors, tensors</span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="-webkit-text-stroke-width: 0px; float: none; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; widows: 2; word-spacing: 0px;"> etc. are very important to understand relativity and
cosmology. So, I will be writing about these gems, very soon…</span></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="-webkit-text-stroke-width: 0px; float: none; font-variant-caps: normal; font-variant-ligatures: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-decoration-thickness: initial; widows: 2; word-spacing: 0px;">Once these articles will be available links will be provided here...</span></span></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and <b>follow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar if you are viewing in desktop and if you are viewing from mobile then its in the bottom of the page).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><br /></div><div class="MsoNormal" style="text-align: justify;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 19.26px;"></span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px;"></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 19.26px; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div><div style="margin: 0px;"><div><div style="margin: 0px; text-align: justify;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>To know my other works visit my site - <a href="https://ratnadeepdchoudhury.com/" rel="" target="_blank"><b>ratnadeepdchoudhury.com</b></a></i></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: justify; vertical-align: baseline;"></div></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; text-align: left; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394; font-size: 11pt; line-height: 15.6933px;">To meet more physics Enthusiasts please join our</span><span style="font-size: 11pt; line-height: 15.6933px;"> </span></i><span style="font-size: 11pt; line-height: 15.6933px;"><a href="https://www.facebook.com/The-Dynamic-Frequency-452248138674077/?modal=admin_todo_tour"><b><i>Facebook Page</i></b><i> </i></a></span></span></div></div><div style="font-family: lato, sans-serif; text-align: left;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i style="text-align: justify;"><span style="color: #0b5394;">Also for the latest updates of my posts join me on </span><b><a href="https://twitter.com/TheDynamicFreq1">Twitter</a></b></i><span style="color: #0b5394; font-size: 13.5pt;"></span></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i>I also have a youtube channel where I make videos on Productivity & Growth, Physics and Technology. Check it now on <a href="https://www.youtube.com/channel/UC3wGmqgSfHg_J-bsl1xXbng" target="_blank"><b>YouTube</b></a></i></span></div><div style="margin: 0px;"><span style="color: #0b5394; font-family: times, times new roman, serif;"><i><br /></i></span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-4066170427961759562020-12-27T00:32:00.012+05:302021-02-07T13:14:16.510+05:30A Crisped Introduction of Spacetime!!!<div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWKd45mQA7A8RuGIA_Gr4dp7r-9IeHTGEK9TSXzib11JaT9eUFIr2G6aCElqTFqNawT48kXbasweEAVuxmRAadnoQh4b3rBjRC6qYcfgZK_8YZyxhH7y6ITM5e6K2G2b4ibRyesjmWvY-O/s1280/wormhole-2514312_1280.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWKd45mQA7A8RuGIA_Gr4dp7r-9IeHTGEK9TSXzib11JaT9eUFIr2G6aCElqTFqNawT48kXbasweEAVuxmRAadnoQh4b3rBjRC6qYcfgZK_8YZyxhH7y6ITM5e6K2G2b4ibRyesjmWvY-O/s320/wormhole-2514312_1280.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: left;"><br /></div></div><p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;">Space-Time… I am pretty sure if you have ever watched a ‘Space-Show’
or a ‘Sci-fi film’, then there is a high chance that you have heard this word.</span><p></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Its sounds cool, isn’t? … And strangely, it’s concept is also cool,
enough to give you a good adrenaline rush to study astrophysics and its related
mathematics.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Okay… enough of the trailer… let’s dive into the movie and enjoy the
climax!!!<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Space-time, as it sounds, made up of two critical words, <b>space</b> and
<b>time</b>. <b>Space</b>, as per the traditional physics, <b>represents the 3-dimensional co-ordinate
system of our universe.</b> Scientists before Einstein <b>believed that our universe
is limited to these 3 coordinates (x, y and z axes) </b>and the universe according to
this convention is called <b>Newtonian</b>. But, in 1905, E<b>instein showed in his
Special Theory of relativity that time is also an integral factor</b>. Well, it was
not before 1908, when one of the Einstein’s own professor and renowned physicist
<b>Herman Minkowski</b> proposed the concept of the union of space and time as a way to reformulate
Special Relativity and in this way the concept of space-time was born.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Well, before going further deep into this, one concept should be
cleared. In a <b>Newtonian Universe, the laws of physics are the same in all inertial frames
and time is absolute. Also, a coordinate is totally dependent on the 3-axes</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Things get a little bit weird in <b>Minkowski universe (now commonly
known as Minkowski Space)</b>, where the concept of spacetime comes into the play.
Here, the whole universe is represented a <b>4-dimensional coordinated system</b>,
which comprises of the <b>traditional coordinate axes (x, y, z axes) and time</b>. But
this concept comes with a critical catch. This time the time is not absolute as
it was in<b> Newtonian Universe</b>. That means that for different objects the time is
running at different speeds depending on a<b> reference frame</b>. I know its hard to
believe as we are not habituated to this type of concept. When we discuss,
space-time, Minkowski universe and special relativity, the concept of the frame of
reference become a critical thing. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>The frame of reference</b> refers to a point/object, and with respect to this
point/object, we do our calculations and observations. In the space-time concept, it
becomes critical because here quantities like velocities, time, mass, all are relative
to the observer or frame of reference. Only speed of light is the same in all
inertial frames. Minkowski Spacetime and special relativity obey the <b>Lorentz transformations</b>
which were formulated by <b>H.A. Lorentz</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The concept of Space-time is again used in <b>General Theory of
relativity (1916)</b> and this <b>time is used to describe the gravity</b>. Now after the introduction of this theorem, the gravity is no longer a force as it was described in <b>Newtonian
physics rather it’s the warping of space-time</b>. To understand this, imagine that<b>
ball on stretched tarpaulin</b>, now you should expect a curvature around the ball
and the intensity of curvature is directly proportional to the weight of the
ball. A similar case is with the <b>Spacetime fabric</b>. Einstein described gravity
like this way and he is pretty successful with this as the equations of General
relativity <b>finally solved the mysterious irregular orbit of planet Mercury</b>. But,
till this date no one knows from where this spacetime fabric was created and
why it actually bends, we only know the effects not the reason behind it. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Well, there is a lot in the story but they are outside the scope of
this article and I am planning to write those in my upcoming articles. Topics
like <b><a href="https://thedynamicfrequency.blogspot.com/2021/01/lorentz-transformation-simplified.html" target="_blank">Lorentz Transformation</a>, <a href="https://thedynamicfrequency.blogspot.com/2021/01/spacetime-interval-invariant.html" target="_blank">spacetime interval and how its invariant</a>, 4-vectors,
tensors</b> etc. are very important to understand relativity and cosmology. So, I
will be writing about these gems, very soon…<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>And a very happy new year 2021 to you and your family. This will be the last article of
this year as well as this decade. Thanks for all the love and support…. See you
in next year…</b></span><o:p></o:p></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of the quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><br /></div><div class="MsoNormal" style="text-align: justify;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 19.26px;"></span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px;"></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 19.26px; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; 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font-family: times, times new roman, serif;"><i>To know my other works visit my site - <a href="https://ratnadeepdchoudhury.com/" rel="" target="_blank"><b>ratnadeepdchoudhury.com</b></a></i></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><o:p></o:p></span></div></div><div style="font-family: lato, sans-serif; margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-33681927340814424562020-11-29T00:26:00.008+05:302020-11-29T00:56:47.209+05:30Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjH-pfNmndsRLXDipcD1BDbdipeFx7SfdpfrmWDzzZeaKIDza6-SAuIsPztFJyFXMQERtxDr2XBtmEpxo6w4gH8Ozyyeel0UUk77J0jnctDo_TacJXfhVqTAzAP0xWMsC446vnjrqJkjthS/s1280/satellite-5181683_1280.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjH-pfNmndsRLXDipcD1BDbdipeFx7SfdpfrmWDzzZeaKIDza6-SAuIsPztFJyFXMQERtxDr2XBtmEpxo6w4gH8Ozyyeel0UUk77J0jnctDo_TacJXfhVqTAzAP0xWMsC446vnjrqJkjthS/s320/satellite-5181683_1280.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><p></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Well, when
old-school telecom companies are pushing the limits of ground-based telecom
services by doing extensive research on the technologies like <b>5G</b>, there are
some <b>newcomers in the industry</b>, trying their hands in a new direction.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><b>SpaceX
backed Starlink, Amazon backed Project Kuiper and private companies like OneWeb</b>
are the leaders in this new approach. They are trying to provide internet, even
to remote places on the earth by using Satellites. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">In past 20
years, <b>the cost for launching satellites by using rockets had been reduced
significantly</b>, so much so that even after providing internet by using satellites,
these companies can make <b>profit</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Satellite
Internet system will surely <b>help those countries which have poor infrastructure
and also can provide internet to remote locations on earth in a affordable way</b>.
<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">According
to the facts, SpaceX’s Starlink is currently the market leader in this segment with
currently <b>904 satellites</b> in orbit (as of 25th November, 2020). <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Next to
the competition, we have OneWeb with <b>74 satellites launched till date</b> (as of 28<sup>th</sup>
November, 2020).<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">And at
last we have Project Kuiper of Amazon. Till date, it's only a planned program of
Amazon and no satellites had been launched. Amazon announced that it will invest
more than <b>10 billion USD in Project Kuiper</b>. For this, <b>Amazon also hired ex-SpaceX’s
Starlink Vice President Rajeev Badyal and appointed him as the president of
Project Kuiper</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">So let’s dig
into the details of each company, one by one….<o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>1) SpaceX’s
Starlink:</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b><br /></b></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="color: red; font-size: x-large;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9EcH8DZfk-Q_34JJyV0KvKh-aQ5aJWpCoKmsp2COy4c37O_dTba3xU4beXHakBg1o3CZcxFxTw0DVzn_Q-FKrvcjqyJYOEjdj7KTAgewlRLi_OrMvJLSWBfg4emzoLTuUlERy9DhFv47F/s250/250px-Starlink_Logo.svg.png" style="margin-left: 1em; margin-right: 1em;"><img alt="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" border="0" data-original-height="127" data-original-width="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9EcH8DZfk-Q_34JJyV0KvKh-aQ5aJWpCoKmsp2COy4c37O_dTba3xU4beXHakBg1o3CZcxFxTw0DVzn_Q-FKrvcjqyJYOEjdj7KTAgewlRLi_OrMvJLSWBfg4emzoLTuUlERy9DhFv47F/s16000/250px-Starlink_Logo.svg.png" title="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" /></a></span></div><p></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Among
these three competitors, Starlink is most progressive. Ranging from<b><a href="https://thedynamicfrequency.blogspot.com/2020/09/space-x-dragon-engineering.html" target="_blank"> SpaceX’s Dragon and Crew Dragon program</a></b> and <b><a href="https://thedynamicfrequency.blogspot.com/2019/11/spacex-starship-BFR-details.html" target="_blank">Big Falcon Rocket</a></b> to this, 2020 was a great year for SpaceX and
for Elon Musk also. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">All of the
SpaceX’s satellites are in <b>Low Earth orbit</b>. Till now they have<b> launched 955
satellites but only 904 are operational</b>. They have planned to launch more
satellites in coming years. All of their satellites were launched by SpaceX’s
Falcon 9 rocket and except the first launch, when they used Falcon 9 Full
Throttle (FT), in every other launch they used Falcon 9 Block-5.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Speaking
about Satellite Hardware of Starlink, they all are SmallSat class satellites
and have weight of approximately <b>100 to 500 kgs</b>. All are in LEO orbit (Low-earth
orbit). These satellites will employ <b>optical inter-satellite links and phased
array beam</b>. These satellites will provide signals in <b>Ku (12 to 18 Ghz) and Ka (26.5
to 40 Ghz) bands</b> of the electromagnetic spectrum. Phased array beams refer to
the computer-controlled array of antennas which creates a beam of
electromagnetic beams and those beams can be steered electronically without
moving the antennas.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">A minimum
theoretical <b>round-trip latency has been assumed to be 25 to 35 milliseconds</b> as
these satellites will orbit at the height of <b>1/105 to 1/30 that of
geostationary orbits</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The
satellites will also use <b>Hall effect Thrusters which are a type of ion thruster
in which the propellent is accelerated by an electric field</b>. These Hall effect
thrusters will be used with <b>krypton gas as the reaction mass</b> as it tends to
exhibit significantly higher erosion of the flow channel compared to a similar
to a system operated with Xenon. In my subsequent articles, I will explain about
Hall effect thrusters.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Speaking
about applications, do not expect that these smallsats will be used for
satellite phones like <b>Globalstar </b>or<b> Iridium</b>. Rather, it will use <b>terminals
which will have phased array antennas</b> and will track the satellites and thus
transfer the data.<o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>2) OneWeb:</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlVT6RfMpaECttlpUGTmZkEKNTawM9jDCG77yhXqIx7TX98sBW_hhmeiTqaEOHaxlBV6SmplDQQHCPpVMxiKj2nou7DOipzSWcd2GmRwqp6wEn3XfJKIWaxarbFlE3AYY9NasrT2Nmt6UZ/s250/250px-OneWeb_Logo.png" style="margin-left: 1em; margin-right: 1em;"><img alt="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" border="0" data-original-height="76" data-original-width="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlVT6RfMpaECttlpUGTmZkEKNTawM9jDCG77yhXqIx7TX98sBW_hhmeiTqaEOHaxlBV6SmplDQQHCPpVMxiKj2nou7DOipzSWcd2GmRwqp6wEn3XfJKIWaxarbFlE3AYY9NasrT2Nmt6UZ/s16000/250px-OneWeb_Logo.png" title="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" /></a></div><br /><span style="color: #0b5394;"><br /></span><p></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">OneWeb is
a global communication provider company whose main purpose is to provide satellite-based
internet services to the world. This company is headquartered in <b>London, United
Kingdom</b> and the current <b>owners</b> of the company are <b>Government of United Kingdom</b>
and <b>Bharti Enterprises (that same group which owns Airtel Telecommunications)</b>.
Both the <b>owners have 42.2 % stake</b> in the company. This company was <b>founded in
2012 by Greg Wyler</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Unlike
Starlink, this company have troubled history due to COVID-19 and 2020 Stock Market
Crash. So much so, that they have to file bankruptcy and later saved by joint
venture of UK govt. and Bharti group by providing an<b> investment of 500 million USD by each of the current owners</b>. They <b>started launching satellites in February 2019</b> and have <b>planned to launch
more than 648 LEO (Low Earth Orbit)</b> satellites. By March 2020, they had launched
74 satellites. Their all satellites had been launched by <b>Soyuz rockets of Russia</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Speaking
about the satellite hardware and configuration, the OneWeb Constellation
satellites have approximately <b>150kg in mass</b>. The <b>648 operational satellites are
to be operate in 18 polar orbits and will have an orbital height of 1200 km
above sea level</b>. Like Starlink, these satellites will also <b>provide Ku Band</b> of
electromagnetic Spectrum. Also, like Starlink,<b> phased antennas will be used to communication
purposes and can provide internet access at 50 megabits/seconds</b>. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>3) Amazon’s
Project Kuiper:</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJKj2BvQmN8u_6T9ll97-MjCCj78ziklVYWeA-h3mgE3-hoZ3_M7ATIUZWLkm-gljCpOxxmy0pbyQu-i6DOn_k-uMnBdRX85Cj1hwu1iS7IhwanmrUBDd34XDfBY4cEkIDK0RWV3U_lYjn/s220/220px-Project_Kuiper.jpg" style="margin-left: 1em; margin-right: 1em;"><img alt="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" border="0" data-original-height="52" data-original-width="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJKj2BvQmN8u_6T9ll97-MjCCj78ziklVYWeA-h3mgE3-hoZ3_M7ATIUZWLkm-gljCpOxxmy0pbyQu-i6DOn_k-uMnBdRX85Cj1hwu1iS7IhwanmrUBDd34XDfBY4cEkIDK0RWV3U_lYjn/s16000/220px-Project_Kuiper.jpg" title="Satellite Internet Wars: Starlink vs OneWeb vs Project Kuiper The Dynamic Frequency Ratnadeep Das Choudhury" /></a></div><br /><span style="color: #0b5394;"><br /></span><p></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Amazon
announced about Project Kuiper in <b>April 2019 </b>and they are expecting to launch a
massive <b>3236 satellites to achieve their goal</b>. According to Amazon, it would take
a decade to launch full constellation.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Till date,
<b>they haven’t launched a single satellite</b>. Their satellites are projected to orbit
with an altitude of <b>590 to 630 km</b>. These satellites will operate in <b>98 orbital
planes in 3 orbital shells, one each at 590 km , 610 km and 630 km</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p><span style="color: #0b5394;"> </span></o:p></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Major Concerns Regarding Satellite Internet Concept:</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">As these projects need to have large number of satellites, these satellites
add to already increasing <b>Space debris problem</b> in our orbits and thus risk of
causing satellite collision is increasing.<o:p></o:p></span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRQ4GirckZq27mzwcroby-TvEIPwXftGFTKYTariyoouHoEXhCpK4xH9bxvK0gCyiewmTafchP92p4KJ6k1H-fNHE-qD2dNxaGDgEj12CRZAAU__emTP5_Nnc23-lNBD2YUZyCRKRIQxo6/s236/download.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Disruption caused by Starlink Satellite imaging through a telescope The Dynamic Frequency Ratnadeep Das Choudhury" border="0" data-original-height="214" data-original-width="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRQ4GirckZq27mzwcroby-TvEIPwXftGFTKYTariyoouHoEXhCpK4xH9bxvK0gCyiewmTafchP92p4KJ6k1H-fNHE-qD2dNxaGDgEj12CRZAAU__emTP5_Nnc23-lNBD2YUZyCRKRIQxo6/s16000/download.jpg" title="Disruption caused by Starlink Satellite imaging through a telescopeThe Dynamic Frequency Ratnadeep Das Choudhury" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Disruption caused by Starlink Satellite imaging through a telescope </td></tr></tbody></table><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394; font-size: 13.5pt;"><br /></span></span><p></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394; font-size: 13.5pt;">Along with this, the large number of satellites also </span><b style="color: #0b5394; font-size: 13.5pt;">interfere with
astronomical observations</b><span style="color: #0b5394; font-size: 13.5pt;">. </span><b style="color: #0b5394; font-size: 13.5pt;">The International Astronomical Union (IAU) and National
Radio Astronomy Observatory (NRAO) already raised concern about these</b><span style="color: #0b5394; font-size: 13.5pt;">.</span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">No doubt, these technologies will help out society to upgrade to the
next level and also help people living in the remotest areas to access the internet
but, if they can tackle the concerns raised against them and will not hamper
the operations of our astronomers then only this technology will be successful…Lets
see what the future stores for us…</span><o:p></o:p></span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"></div><div class="MsoNormal" style="background-color: white; color: red; font-family: lato, sans-serif; font-size: 15px; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"></span></span><br /><br /></div><div class="MsoNormal" style="text-align: justify;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 19.26px;"></span></div><p class="MsoNormal"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 19.26px;"></span></p><p class="MsoNormal"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"></span></p><div class="MsoNormal" style="background-color: white; font-family: lato, sans-serif; line-height: normal; margin-bottom: 0.0001pt; text-align: justify; vertical-align: baseline;"><div style="font-family: "Times New Roman"; margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;">Thanks for Reading!!!</span></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Check out my other blog posts!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Don't forget to share this post in your social media handles to enrich everyone's knowledge!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><br /></div></div><div style="margin: 0cm 0cm 0.0001pt; vertical-align: baseline;"><div style="margin: 0px;"><span style="font-family: "times" , "times new roman" , serif;"><i><span style="color: #0b5394;">You can directly talk with me on</span> <b><a href="https://www.instagram.com/the_dynamic_frequency/?hl=en">Instagram</a></b></i><o:p></o:p></span></div></div><div style="margin: 0cm 0cm 0.0001pt; 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vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com1tag:blogger.com,1999:blog-3578987301814134971.post-5642244237709099762020-10-17T23:51:00.015+05:302021-03-26T09:33:47.625+05:30Quantum Harmonic Oscillator Part-1: Introduction in a Nutshell<p><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;"></span></p><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: left;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4xYBZuzO_mokPIpy6vWsBTCioo7i8gfA23rqLfUhQgnVVxeEu7XDo-_XY5iYSmh_FVvAzfZzFyehHu10YNIjbrjwiPtbco3OB8jGOT9jeGlfUlMH9aEJR92v_WkBdlvQo79XGJHT21San/s799/6th_Eigenfunction_of_the_2D_Simple_Harmonic_Oscillator_2nd_perspective_view.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="509" data-original-width="799" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4xYBZuzO_mokPIpy6vWsBTCioo7i8gfA23rqLfUhQgnVVxeEu7XDo-_XY5iYSmh_FVvAzfZzFyehHu10YNIjbrjwiPtbco3OB8jGOT9jeGlfUlMH9aEJR92v_WkBdlvQo79XGJHT21San/s320/6th_Eigenfunction_of_the_2D_Simple_Harmonic_Oscillator_2nd_perspective_view.jpg" width="320" /></a></div><br /><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: justify;"><br /></span></div><div class="separator" style="clear: both; text-align: left;"><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: justify;">First of all, don’t be afraid of the title, scientists love to use
fancy words to describe even a simple topic and why wouldn’t they, it makes
physics a lot more appealing.</span></div></div>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Well, Quantum Harmonic Oscillator is one such topic. Fancy name with
fancy application.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>The goal of this article is to give a solid foundation in this topic</b>.
I will try my best to make you understand this concept and give you the essence
of it rather than only spitting out some random equations and their derivations
like most textbooks and online sites do, after-all, this the motto of our
website.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">So, let’s start….</span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">To understand the Quantum Harmonic Oscillator, firstly we have to have
an idea on the <b>classical harmonic oscillator</b> and the<b> basic meaning of Harmonic
motion</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Harmonic Motion and Classical Harmonic Oscillator</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The harmonic motion refers to<b> the periodic motion or periodic vibration.</b> To
understand it better, just imagine a string of a guitar is vibrating, then those
motions are symmetrical about a region of <b>equilibrium and distance of each trough
and crest from the median line is same when a continuous force of same amount is
applied</b>. This type of motions may have a single frequency or amplitude or may
be composed of two or more such simple type of motions.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;"><b>Classical Harmonic Oscillator refers to a system which if displaced
from the equilibrium, experiences a force named as restoring force</b>. This force
is proportional to the displace x. The whole system can be imagined like an
object with mass m is attached with the spring, and when the spring vibrates it
created a graph with<b> uniform tough and crest</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">If that force is the only force acting on the system then the system
is called as S<b>imple Harmonic Oscillator</b> and the motion it shows us is called
Simple Harmonic motion (SFM) and the equation of force in that case is</span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;"><b>F
= </b></span></span><span style="color: red;"><b><span style="font-family: "Courier New"; font-size: 13.5pt; line-height: 107%;">-</span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;">kx<span style="mso-spacerun: yes;"> </span>(</span><span style="font-family: "Courier New"; font-size: 13.5pt; line-height: 107%;">-</span></b></span><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: red;"><b>∞ ≤ x ≥ ∞ )<span style="mso-spacerun: yes;"> </span>(1)</b></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Where, F and X are vectors and k (spring constant) is a positive
constant<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Here in the above equation, the negative sign refers to the <b>opposite
direction of the motion as compared to the direction of the force applied</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Before starting our discussion about the quantum harmonic oscillator,
we need to know a couple of more formulae and concepts…<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">The <b>time period (period after which the motion repeats itself)</b> for the
simple harmonic motion is <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span><span style="mso-tab-count: 2;"><span style="color: #0b5394;"> </span><span style="color: red;"><b> </b></span></span><span style="color: red;"><b>T
= 2π√(m/k)<span style="mso-spacerun: yes;"> </span>(2)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Where, m is the mass of the object undergoing the SHM and k is the
spring constant<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">One thing which we should keep in mind that SHM is not damped motion.
<b>A damped motion is such kind of motion which dies after a period of time. In the reality most of the motions and oscillations are damped unless and until they
are given some sought of force or energy</b>. Discussing further about the
formulation and derivations on the damping motion is beyond the scope of this
article. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Another formula related to classical harmonic oscillator is the
solution of the differential equation of (1),</span></span></p><p class="MsoNormal" style="tab-stops: 151.5pt;"><b style="text-align: center; text-indent: 151.5pt;"><span style="color: red;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"> x(t) = A cos(</span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">t + </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">φ</span></span></b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%; text-align: center; text-indent: 151.5pt;"><b><span style="color: red;">)<span style="mso-spacerun: yes;">
</span>(3)</span></b></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">where, A is amplitude of the wave, </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> is the angular
frequency and </span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">φ</span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"> is the phase.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="color: #0b5394;"><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">Here,<b> </b></span><b><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;">ω </span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">= </span></b><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><b>√(k/m)</b> and thus eq. (2) can be written as <b>T= 2π/</b></span><span style="background: white; font-family: Cambria, serif; font-size: 13.5pt; line-height: 107%;"><b> ω</b></span><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt;"><span style="font-family: "Footlight MT Light",serif; font-size: 13.5pt; line-height: 107%; mso-bidi-font-family: Cambria;"><span style="color: #0b5394;">Also, to get into the <span style="background: white;">Schrodinger’s
equation and related it to harmonic oscillation, we need to get the expression
for potential, so let’s get it,<o:p></o:p></span></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span><span style="color: red;"><b><span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">V(x) = </span><span style="font-family: "Courier New"; font-size: 13.5pt; line-height: 107%;">-</span><span face=""Nirmala UI", sans-serif" style="font-size: 13.5pt; line-height: 107%;">ഽ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">F dx
= </span><span face=""Nirmala UI", sans-serif" style="font-size: 13.5pt; line-height: 107%;">ഽ</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">kx dx
= kx</span></b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>²/2 + c<span style="mso-spacerun: yes;"> </span>(4)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Where, c
is the constant of integration<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Note here
the use of negative during integration. Here, its use to denote the opposite
direction of potential energy stored as compared to the direction of force
applied. <o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Equation 4
is also named as Harmonic oscillator potential.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><b>The value
of c is normally dropped in the further calculation for the sake of simplicity
and also because it doesn’t change the final value drastically</b>.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Quantum
Harmonic Oscillator</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Here we
go, finally in the fun part….<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">As you
know, <b>Quantum Harmonic Oscillator (QHO)</b> is a very vast topic. In fact, it’s one
of the few topics in QM for which an analytic and exact solution is known to
us. That makes this topic one of the key systems to study QM.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">QHO is the
quantum mechanical version of classical harmonic oscillator and that’s why I
preferred to give a brief intro to the topic.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">So, lets
start from <b>forming <a href="https://thedynamicfrequency.blogspot.com/2019/10/schrodingers-time-dependenteq.html" target="_blank">Schrodinger’s equation</a></b>,<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">So, by
putting equation 4 in Schrodinger’s equation, we get,<o:p></o:p></span></span></p>
<p align="center" class="MsoNormal" style="tab-stops: 151.5pt; text-align: center;"><span><span style="background: white; font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"><span style="mso-spacerun: yes;"><span style="color: #0b5394;"> </span></span><span style="color: red;"><b>-
(ħ<span style="font-family: "Footlight MT Light", serif;">²</span>/2m)(d</b></span></span><span style="color: red;"><b><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"><span style="font-family: "Footlight MT Light", serif;">²</span> 𝜓ₙ/dx<span style="font-family: "Footlight MT Light", serif;">²</span>) + </span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">kx</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;">²/2 </span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">𝜓ₙ = </span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">E</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;">ₙ</span><span style="font-family: "Cambria Math", serif; font-size: 13.5pt; line-height: 107%;"> 𝜓ₙ<span style="mso-spacerun: yes;"> </span></span></b></span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>(5)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">This equation will be used in my future articles on this topic.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Since I
am intended to write this article as an introductory article to QHO, so <b>I will
not go further deep into the formulation on this topic, rather I like to give you
a brief overview of it. In future posts</b>, I will tackle each and every
necessary topic under QHO.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Now, here
I want to introduce you the <b>applications of Quantum Harmonic Oscillator theory</b>:<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">1) It can
be used to describe the vibration spectra of diatomic molecules and complex
modes of vibration in larger molecules<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">2) It is
used in the theory of vibrational heat capacity<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">3) I can
be used to understand the motion of atoms in solid lattice and explains who
phonons arise from them. In the periodic and elastic arrangement of atoms or
molecules of condensed matter (solids and some special liquids), if any
collective excitation happens then that called as a Phonon. Using QHO, we can
find various properties regarding that, for example, we can find the exact amount of
energy, which if supplied to the QHO lattice then it can push it to the next
energy level.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">4) It can
also, be useful to find the expression for thermodynamic vibrational energy.<o:p></o:p></span></span></p>
<p class="MsoNormal" style="tab-stops: 151.5pt; text-align: justify;"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">In my
further articles, I will try to tackle each and every necessary topic which are
needed to understand QHO. In <b>part 2</b> of <b>QHO series</b>, I will explain <b><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank">Schr</a></b></span></span><span style="color: #0b5394;"><b><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank"><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt; text-align: left;">ödinger's equation with d</span><span style="font-family: "Footlight MT Light", serif; font-size: 13.5pt;">imensionless</span></a> </b></span><span style="color: #0b5394; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><b><a href="https://thedynamicfrequency.blogspot.com/2021/02/quantum-harmonic-oscillator-part2.html" target="_blank">terms</a></b>. On the basis of that article, I will try to explain the <b><a href="https://thedynamicfrequency.blogspot.com/2021/03/quantumharmonic-part3-asymptoticSol.html" target="_blank">Asymptotic Solution of QHO</a></b>. Once they are uploaded, the links will be provided here.</span></p><div style="margin: 0px;"><span style="color: #0b5394;"><span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 19.26px;">If you like my articles then please <b>consider subscribing our newsletter</b> and f<b>ollow this blog</b> to get the <b>latest updates directly to you</b> <o:p></o:p></span><b style="font-family: "footlight mt light", serif; font-size: 18px; outline: 0px;">(buttons are available in the right sidebar).</b></span></div><div class="MsoNormal" style="text-align: justify;"></div><div class="MsoNormal" style="text-align: justify;"><br /></div><div style="margin: 0px;"><div style="margin: 0px;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b style="color: #474747;">To know basics of quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" target="_blank">"Through the wormhole" on amazon kindle.</a></b></span></span></div></div><div style="margin: 0px;"></div><div class="MsoNormal" style="background-color: white; 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vertical-align: baseline;"><div style="margin: 0px;"><i><br /></i></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">See you Again!!!!!</span></div></div><div style="font-family: "Times New Roman"; text-align: left;"><div style="margin: 0px;"><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> -Ratnadeep Das Choudhury</span></div></div><div style="font-family: "times new roman";"><div style="margin: 0px;"><span style="color: lime; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"> Founder and writer of The Dynamic Frequency</span></div></div></div>Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-49038321047494165192020-09-25T02:05:00.000+05:302020-09-25T19:09:04.943+05:30LightSail-2: Miracle of Solar radiation propulsion!!!<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglFGlL4sSmPbh12V-hUb7PQ1XT3dpwF33XqVXbEnw9V2cc3dWalJqWPE6Co-ViNFsgisQip4LDOAXhegRvxLmubtrB3pYru0SJXvMDHDd-SFk35bbOEGpFbaLc0sUDj7IHzqYJf0TTq1ZC/s1600/LightSail_2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="900" data-original-width="1600" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglFGlL4sSmPbh12V-hUb7PQ1XT3dpwF33XqVXbEnw9V2cc3dWalJqWPE6Co-ViNFsgisQip4LDOAXhegRvxLmubtrB3pYru0SJXvMDHDd-SFk35bbOEGpFbaLc0sUDj7IHzqYJf0TTq1ZC/s640/LightSail_2.jpg" title="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" width="640" /></a></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Well, once in a while, some amazing humans come forward with their
amazing inventions which have the potential to improve human technology
drastically. <b>LightSail </b>is such a technology!!!<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">LightSail is a <b>crowdfunded project</b> of <b>the Planetary Society</b>, a
<b>non-profit organization</b>. This project was started to test the <b>solar sailing
technology</b>. This technology primarily focused on <b>CubeSats</b>. They decided to make
this technology for CubeSats because they have <b>revolutionized the space
industry mainly because of tech miniaturization</b> and at the same time carry<b> good
payloads</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The concept of solar sailing was first imagined by <b>Johannes Kepler
over 400 years ago</b>. It was later embraced by famous scientists like <b>Carl Sagan,
Bruce Murray and Louis Friedman</b>. <o:p></o:p></span></span></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9NiPCUxIifN77bsHHgZuicCuTeayhTo0PK8gWDd6VkWZRG9pgRciV9snbRVWLDlW4ppZHZ-FGobpTixQvS-viN6fsAfi7NGGsQl6_p3YdUOvslU02GpufaOcvBuQGlnlD2KNKBcyBtWoF/s1600/Tycho-Brahe-244x300.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="300" data-original-width="244" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9NiPCUxIifN77bsHHgZuicCuTeayhTo0PK8gWDd6VkWZRG9pgRciV9snbRVWLDlW4ppZHZ-FGobpTixQvS-viN6fsAfi7NGGsQl6_p3YdUOvslU02GpufaOcvBuQGlnlD2KNKBcyBtWoF/s1600/Tycho-Brahe-244x300.jpg" title="Johannes Kepler | The dynamic frequency Ratnadeep Das Choudhury" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Johannes Kepler</td></tr>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">LightSail project consists of two spacecraft namely <b>LightSail-1</b> and
<b>LightSail-2</b>. This article will be more focussed on LightSail-2 as it’s the most
recent and most influential spacecraft of LightSail project.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">But before understanding LightSail-2, firstly, we have to know the
basics of LightSail-1 spacecraft and its mission. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><b>LightSail-1 (formerly LightSail-A)</b> was more like a<b> technological and
engineering sample</b> spacecraft. It <b>didn’t perform solar sailing</b>. It was launched
by <b>Atlas V</b> rocket on 20<sup>th</sup> May 2015. During its mission, t<b>he mission-control
lost the communications with the spacecraft for two times</b>. Later the communications
were re-established and <b>on 10<sup>th</sup> June 2015 the mission was declared
a success</b>. It re-entered the atmosphere on 14<sup>th</sup> June 2015.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy8PYn1fyrOYYLrWe4F2ZWKPz54ZGYUN14ztwodoiXcSJ7Wh_e6iJMLweU9hs-lmaIEIGXugIl6RdOK-EgnKhl08EhCreym7WFySRJ_-Gz_T9k6ZC95Ws8dV7z_TnyFLZlSStS5_aLaqx-/s1600/LightSail_1_with_deployed_solar_sail.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy8PYn1fyrOYYLrWe4F2ZWKPz54ZGYUN14ztwodoiXcSJ7Wh_e6iJMLweU9hs-lmaIEIGXugIl6RdOK-EgnKhl08EhCreym7WFySRJ_-Gz_T9k6ZC95Ws8dV7z_TnyFLZlSStS5_aLaqx-/s320/LightSail_1_with_deployed_solar_sail.jpg" title="LightSail-1's successful deployment | The dynamic frequency Ratnadeep Das Choudhury" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">LightSail-1's successful deployment</td></tr>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br />After the success of LightSail-1, Planetary Society’s next target was
<b>LightSail-2</b> and this time they intended <b>to perform actual solar sailing</b>. After
learning from all the mistakes, they did in their first mission, they were
ready to launch it. So, it was launched on <b>25<sup>th</sup> June 2019 by SpaceX’s
Falcon Heavy</b>. LightSail-2 was enclosed within a <b>Georgia Tech student-built
spacecraft</b> named as <b>Prox-1</b>. Prox-1 detached from the Falcon heavy into a
circular,<b> 720 km high orbit </b>and successfully deployed LightSail-2. <b>This orbit
is much higher than that of LightSail-1</b>. After deployment, it opened its hinged
solar arrays on 23<sup>rd</sup> July 2019. <o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKYHhDft5caCh26tlxwh7HQRROX7cvJRMkQOkj6fTgyN7wz3uwfDY3V2MrnrzE_eo6DQNNmxxMhKys4goiDiOkpBkaqwrDP0r6bsYNQ_pvYpyG1rKNTVskNEqTmo9NTmbv4-etQecB3MzW/s1600/Lightsail2_AutoA.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="1103" data-original-width="1600" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKYHhDft5caCh26tlxwh7HQRROX7cvJRMkQOkj6fTgyN7wz3uwfDY3V2MrnrzE_eo6DQNNmxxMhKys4goiDiOkpBkaqwrDP0r6bsYNQ_pvYpyG1rKNTVskNEqTmo9NTmbv4-etQecB3MzW/s320/Lightsail2_AutoA.jpg" title="SpaceX's Falcon Heavy carrying LightSail-2 (source: NASA) | The dynamic frequency Ratnadeep Das Choudhury" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">SpaceX's Falcon Heavy carrying LightSail-2 (source: NASA)</td></tr>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Once the sail was deployed, the spacecraft was ready to perform its
real task, i.e. solar sailing. The mission objective was to raise its orbit and
for that, <b>the spacecraft had to swing its solar sails into and away from the
sun</b>. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">As it’s a solar sail, <b>the craft’s propulsion relies on solar radiation</b>
and <b>not the charged particles of the solar wind</b>. Since it is using solar radiation
for propulsion, <b>the photons of the light coming from the sun exert radiation
pressure</b> <b>on the sail</b> and thus <b>producing acceleration</b>. Acceleration is <b>relative
to sail’s area to its mass</b>. It means that the larger the area and lower the
mass, the acceleration will be better provided that the spacecraft is getting the same amount of solar radiation. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><b>Radiation pressure</b> is the mechanical pressure exerted by the exchange
of momentum between an object (here, sail) and electromagnetic field.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtVe5r9dyp4pDx-dTK8RPdSIs0Fm6QRydItoPll2tyETtWRJ4TdqD7qJ_7I6CTfETitQ1rVjQH9JenFeYSuZzXoXau13ZjLYPk2ISXAU7E1d9KwFKrJAesy31baPu01YSvIXOzpGJZqtNs/s1600/LightSail2orbitraising.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="270" data-original-width="480" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtVe5r9dyp4pDx-dTK8RPdSIs0Fm6QRydItoPll2tyETtWRJ4TdqD7qJ_7I6CTfETitQ1rVjQH9JenFeYSuZzXoXau13ZjLYPk2ISXAU7E1d9KwFKrJAesy31baPu01YSvIXOzpGJZqtNs/s400/LightSail2orbitraising.gif" title="Orbit raising procedure of LightSail-2 (source: The Planetary Society. Josh Spradling) | The dynamic frequency Ratnadeep Das Choudhury" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Orbit raising procedure of LightSail-2 (source: The Planetary Society. Josh Spradling)</td></tr>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The <b>LightSail-2 sail control algorithm </b>is designed in such a way that
the spacecraft can orient itself and its <b>sail’s edge towards the direction of Sun
when moving towards the Sun</b> and c<b>an orient sail’s face towards Sun when moving away
from the Sun</b>. To orient itself, it uses an<b> internal reaction wheel</b>. As we know,
every action has an equal and opposite reaction, so<b> whenever the wheel rotates</b>,
due <b>to the opposite force the spacecraft also orient itself to fix its orientation
accordingly</b>. This strategy was used to <b>increase orbit apogee</b> and they did it
successfully and t<b>hey announced it on 31<sup>st</sup> July, 2019</b>. But, <b>as the orbit
perigee decays</b>, it will <b>no longer possible to raise the orbit apogee by using solar
sailing</b>.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixhjtw6hq3unPzTB4To4PWAG9VOzkzmFLi24wmTyb9t6E5SsdqJ6qLUbZavx-5wlR4tL7atfBmukSqgdJzsNwYNil2SXVbFC4EsqfyOcoO2fJrXum4VNeVzE9FQLDSv-BOg2u2BmTWYfxc/s1600/50123644692_e009d951a1_b.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="LightSail-2: Miracle of Solar radiation propulsion!!! The dynamic frequency Ratnadeep Das Choudhury" border="0" data-original-height="847" data-original-width="1024" height="264" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixhjtw6hq3unPzTB4To4PWAG9VOzkzmFLi24wmTyb9t6E5SsdqJ6qLUbZavx-5wlR4tL7atfBmukSqgdJzsNwYNil2SXVbFC4EsqfyOcoO2fJrXum4VNeVzE9FQLDSv-BOg2u2BmTWYfxc/s320/50123644692_e009d951a1_b.jpg" title="LightSail-2's Succesful deployment | The dynamic frequency Ratnadeep Das Choudhury" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">LightSail-2's Succesful deployment</td></tr>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The project team initially planned to re-enter Earth’s atmosphere
after approximately one year but thankfully <b>an extended mission was approved on
25<sup>th</sup> Jun, 2020</b>. According to their estimations, LightSail-2’s <b>orbit
decay </b>is around <b>20-34 meters per day</b> and depends upon the level of active sail
control and with this estimation, it is expected that this <b>craft will be in the
orbit until the first or second half of 2021</b>. Its extended mission objectives may include
<b>the characterization of the orbit decay rates</b> and <b>parallelly imaging operation
will continue during the deorbit phase</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Total Cost of LightSail project (both LightSail-1 and LightSail-2) is
about <b>US $ 7 million and it was paid approximately by 40,000 individuals.</b> It is
including<b> US $ 1.24 million </b>raised from a <b>Kickstarter program</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">After, the success of LightSail-2, Planetary Society stated that <b>they currently
have no plans for LightSail-3</b>.</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>LightSail-2 Specifications:</b></span><span style="font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Dimensions when sails stowed:</span></b> <span style="color: #0b5394;">10 x 10 x 30 cm</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Mass:</span></b> <span style="color: #0b5394;">5 Kg</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Sail Thickness:</span></b><span style="color: #0b5394;"> 4.5 micro metre</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Sail material:</span></b> <span style="color: #0b5394;">Mylar, a reflective polyester film</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Sail Width:</span></b> <span style="color: #0b5394;">5.6 m</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Sail Boom length:</span></b> <span style="color: #0b5394;">4 m</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>Total Sail Area:</b></span> <span style="color: #0b5394;">32 m^2 or 340 sq ft</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: red;"><b>Bus (general model of spacecraft):</b></span><span style="color: #0b5394;"> 3U CubeSat</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Sail Layout:</span></b> <span style="color: #0b5394;">4 triangular sails forming a square connected with cobalt
steel booms and booms unwind like tape measures</span><o:p></o:p></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">LightSail has created new possibilities in the field of solar
propulsion. This type of innovations can be used later to develop more complex satellites
which will eventually help us to conquer new heights in space exploration.</span><o:p></o:p></span></div>
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Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-83824340776337279662020-09-03T00:25:00.000+05:302020-09-03T02:10:57.193+05:30Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!!<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLnEXCuggXB-QSYuKMkUUITfs7phqjVDC3MAdWC-nL_vahJmo4GiO2lRVVQxKoMU2zGkhAWYHUU_W1T_d6qjZHScMrmIAOwi9V1dz5kiR192LswaLBYyMMWNK7637BNpd24buGi1VrHTi_/s1600/8MDikT.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!!" border="0" data-original-height="900" data-original-width="1600" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLnEXCuggXB-QSYuKMkUUITfs7phqjVDC3MAdWC-nL_vahJmo4GiO2lRVVQxKoMU2zGkhAWYHUU_W1T_d6qjZHScMrmIAOwi9V1dz5kiR192LswaLBYyMMWNK7637BNpd24buGi1VrHTi_/s640/8MDikT.jpg" title="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" width="640" /></a></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Well, Dragons are always fascinating…be it in fiction or in modern
times….<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">If anyone would tell somebody back in 2002 that, there would be <b>one
private company in 2020 which would launch humans in the space</b> and<b> safely
return them to earth</b> then, there would be high chance that people would laugh
at that person.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Now fast forward to 2020, that goal is finally achieved. <b>Elon Musk’s
Space X</b> did it in their <b>first-ever try</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Well, <b>Dragon</b> of SpaceX is <b>operating from 2010</b>, and since then indulged
<b>delivering cargo to International space station</b>. But after 2011, when space
shuttle retired America lacked the capability to launch humans to space and
relied only on <b>Russian’s Soyuz program</b>. <o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTLoLGXHvQPfAlJLwruYCwAqe699sPFGl-B-pJKtrcZ0dWAtV9P42Gf2waqddx7GA4E5CSeyb93trhAPR8M10S7OPmWwIN8I4yLDX9r-iChnLEGLtj6zXGgSSXFkjscbYeGuY2vuCAhMwR/s1600/Launch_of_Falcon_9_carrying_CRS-6_Dragon_%252817170624642%2529.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="1600" data-original-width="1067" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTLoLGXHvQPfAlJLwruYCwAqe699sPFGl-B-pJKtrcZ0dWAtV9P42Gf2waqddx7GA4E5CSeyb93trhAPR8M10S7OPmWwIN8I4yLDX9r-iChnLEGLtj6zXGgSSXFkjscbYeGuY2vuCAhMwR/s320/Launch_of_Falcon_9_carrying_CRS-6_Dragon_%252817170624642%2529.jpg" title="Falcon 9 carrying CRS-6 Dragon (Dragon 1 used)" width="213" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Falcon 9 carrying CRS-6 Dragon (Dragon 1 used)</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br />SpaceX used their <b>Dragon 1 to supply cargo to ISS</b> which they <b>retired
on 7<sup>th</sup> March 2020</b>. <b>Dragon 2</b> comes with <b>two variant</b>: one for <b>cargo
missions</b> and another for <b>crewed missions</b>. Crewed version of Dragon 2 has been
developed especially to achieve human spaceflight ability and thus winning the contract for the <b>NASA’s Launch America program</b>. <o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhh4YK94pEbEu79i8zrC2NJfjQ4wECQUHOQg04MJ4PuwhS2PCTZsLE5Rn-R5Me46A0q0D_bLxuhqd1w0nic0ZYLJb6TuVDZ0fioyVBiUadiWlLra0QKv7jxiMCPKA-yBktpPXB18w3bRHYZ/s1600/49953052258_b9a7d6971d_k_0.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="574" data-original-width="985" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhh4YK94pEbEu79i8zrC2NJfjQ4wECQUHOQg04MJ4PuwhS2PCTZsLE5Rn-R5Me46A0q0D_bLxuhqd1w0nic0ZYLJb6TuVDZ0fioyVBiUadiWlLra0QKv7jxiMCPKA-yBktpPXB18w3bRHYZ/s320/49953052258_b9a7d6971d_k_0.jpg" title="Space X Demo-2 launch" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Space X Demo-2 launch (source NASA)</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">On <b>30<sup>th</sup> May 2020</b>, SpaceX <b>launched humans</b> for the <b>first time</b>
under their <b>Demo-2</b> mission. The mission lasted <b>63 days, 23 hours and 25
minutes</b>, while being <b>docked</b> for <b>63 days, 9 hours and 8 minutes</b>. Astronauts <b>Bob
Behnken</b> and <b>Doug Hurley</b> were the astronauts involved in this mission. The craft
was later named as <b>Endeavour</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">All of the Dragons are launched by <b>Falcon 9 rockets</b>.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaJeeuZhk9pVHneHvhbOebvLXsGTucrB5OVCJjCB_McpcviMiZ_Kah0wva6ImCtEAisWpsfnHL37NYHf-ugLhxdXcdOJK7L2hNLBIrtdmwnydjBRRZ80w3lYt1Aj61tRKR1YhYYMH2qQK6/s1600/800px-Crew_Demo-1_Mission_%252846535573034%2529.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="1200" data-original-width="800" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaJeeuZhk9pVHneHvhbOebvLXsGTucrB5OVCJjCB_McpcviMiZ_Kah0wva6ImCtEAisWpsfnHL37NYHf-ugLhxdXcdOJK7L2hNLBIrtdmwnydjBRRZ80w3lYt1Aj61tRKR1YhYYMH2qQK6/s320/800px-Crew_Demo-1_Mission_%252846535573034%2529.jpg" title="Maiden launch of Dragon-2 during Demo-1 mission" width="213" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Maiden launch of Dragon-2 during Demo-1 mission (source wikimedia)</td></tr>
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<span style="font-family: "footlight mt light" , serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Engineering and Design of Dragon Spacecraft</b></span><span style="color: #0b5394; font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Dragon spacecraft uses the <b>conventional space capsule</b> which is also
known as<b> ballistic capsule</b> and also has <b>a cargo carrier trunk</b> at the <b>back of
the craft</b>. The cargo carrier trunk also <b>can be extended</b> and is always
<b>unpressurized</b>. It is also attached to <b>two solar panels</b>.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh0NduynfxIMHApwfUiGSkR1H1OQjXLunGkxtFrwCBnop0h2VvY-w3m-j79x5TeBQCklQoBn0lFT_4fUPVEmA0pb4HjjbmyGaFFIbKQ_-steabkzsQ5sf_AS_dyUSuObS959ccgZATcInn/s1600/cosmos_earth_galaxy_nasa_orbit_satellite_science_solar-1000016.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="800" data-original-width="1200" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh0NduynfxIMHApwfUiGSkR1H1OQjXLunGkxtFrwCBnop0h2VvY-w3m-j79x5TeBQCklQoBn0lFT_4fUPVEmA0pb4HjjbmyGaFFIbKQ_-steabkzsQ5sf_AS_dyUSuObS959ccgZATcInn/s320/cosmos_earth_galaxy_nasa_orbit_satellite_science_solar-1000016.jpg" title="Dragon docked with ISS (the upper docked part is the capsule and the lower part which have solar panels in trunk)" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dragon docked with ISS (the upper docked part is the capsule and the lower part which have solar panels in trunk)</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br />For <b>heat resistant shield</b> the craft uses the variant of <b>NASA’s
Phenolic impregnated carbon ablator material known as PICA-X heat shield</b>. The
Phenolic impregnated carbon ablator is a <b>carbon-fibre material infused in
phenolic resin</b>. It has the advantage of <b>low density</b> and its <b>good ablative ability</b>
making it a useful resource for making heat shield. <b>PICA’s thermal conductivity
is lower than conventional carbon phenolics</b>.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyxjXbncU_-HXlIUAz7IsAJ32WRD_B0VrOj_8Zmv9IO5zlVbIgZdcMpaMYl2ZgSNtg15tb6gQvsRKgBlCzZtj47cCcUYXWeWd0iHefy9p096pUyEeBWXerQwZPEJ5HVUXGsB2t5KC2_Bq-/s1600/download.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="176" data-original-width="287" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyxjXbncU_-HXlIUAz7IsAJ32WRD_B0VrOj_8Zmv9IO5zlVbIgZdcMpaMYl2ZgSNtg15tb6gQvsRKgBlCzZtj47cCcUYXWeWd0iHefy9p096pUyEeBWXerQwZPEJ5HVUXGsB2t5KC2_Bq-/s1600/download.jpg" title="Heat shield of Dragon capsule" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Heat shield of Dragon capsule</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The<b> trunk part</b> of the spacecraft is <b>not reusable</b> but the <b>capsule part</b>
is <b>reusable</b> and can be <b>used multiple times</b>. The trunk part separated from the
capsule when the spacecraft re-enters the Earth’s atmosphere and thus <b>burns up
in the atmosphere</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">As we have seen in many other spacecraft, the SpaceX Dragon also <b>uses
three pairs of triple-processor computers (total six computers)</b> to <b>check each
other’s result</b> to <b>reduce the chances of failure of machinery</b> and achieve a
<b>fault-tolerant design</b>. If anyhow any technical malfunction happens due to
radiation or any other reason, <b>one of the pair will initiate soft reboot</b>. These
<b>6 computers</b> make the <b>main flight computers</b>. Other than these the dragon have
another <b>12 computers to support it</b>. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Dragon uses <b>GLACIER</b> and <b>MERLIN freezer modules</b> for transporting
essential and <b>critical scientific payloads</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">The dragon spacecraft have <b>18 Draco thrusters</b> to navigate itself.
<b>Draco thruster</b> is a <b>hypergolic liquid rocket engine</b>. Hypergolic engines are
those engines which <b>use the propellants which spontaneously ignite when they
come in contact with each other</b>. Draco thrusters can generate <b>400 newtons of
thrust</b>. It uses mixture of <b>monomethyl hydrazine fuel</b> and <b>tetroxide oxidizer</b>.<o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZYmehClkcN4o059HhmaPgv3DkAnCI8z34KzPO3sRvMiWZ4e5AGHbhPvFMsJsx4yNibPjg8tHjXgKo0LV1-QVphQMWMUswB9X_3Esl4WMt2t4YHKitX0OtOVXm4HGhfApPUlTFhGEjpKJ6/s1600/Draco_In-space_Firing.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="600" data-original-width="1280" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZYmehClkcN4o059HhmaPgv3DkAnCI8z34KzPO3sRvMiWZ4e5AGHbhPvFMsJsx4yNibPjg8tHjXgKo0LV1-QVphQMWMUswB9X_3Esl4WMt2t4YHKitX0OtOVXm4HGhfApPUlTFhGEjpKJ6/s320/Draco_In-space_Firing.png" title="Dragon using its Draco thrusters" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dragon using its Draco thrusters (souce wikimedia)</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br />For satellite communication, <b>RS-422(commercial standard)</b> and <b>1553
(Military Standard US) serial I/O</b> is used. It also uses Ethernet communications
for IP addressable standard payload service. The communication system also
provides a command <b>uplink of about 300 kbps</b> and <b>data downlink as 300Mbit/s</b>.
Apart from these, <b>S-band telemetry</b> is also used and<b> fault-tolerance</b> is ensured.
S-band is a designation for a part of the <b>microwave band covering frequencies
from 2 to 4 gigahertz (GHz)</b>. This designation is given by<b> <a href="https://www.ieee.org/">Institute of Electrical and Electronics Engineers (IEEE)</a></b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; line-height: 107%;"><span style="color: red; font-size: x-large;"><b>Dragon’s Specification</b></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Cargo Transport Capacity:</span></b><span style="color: #0b5394;"> 3310 kg<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Pressure vessel payload Volume:</span></b><span style="color: #0b5394;"> 10 cubic meter<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Unpressurized payload volume:</span></b><span style="color: #0b5394;"> 0.1 cubic meter<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Solar arrays power production:</span></b><span style="color: #0b5394;"> 4 kW (peak), 1500 (average) at 120 and
28 direct current voltage<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Unpressurised payload volume:</span></b><span style="color: #0b5394;"> 0.1 cubic meter<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Unpressurised non-recoverable normal trunk:</span></b><span style="color: #0b5394;"> 14 cubic meter payload
volume in the 7ft 7 inches (in length) trunk <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><b><span style="color: red;">Unpressurised non-recoverable extended trunk:</span></b><span style="color: #0b5394;"> 34 cubic meter volume
when trunk extended to 4.3 m (in length) <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">It also supports sensors and s<b>pace apertures up to 3.5 m</b> in <b>diameter</b>
in the <b>trunk section</b>. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;">Apart from all of these, Space X also had plans to use their Dragon
spacecraft in their <b>Mars missions</b>. They named it as ‘<b>Red Dragon</b>’. Later the
<b>project was scrapped</b> as the company increased their focus on the <b><a href="https://thedynamicfrequency.blogspot.com/2019/11/spacex-starship-BFR-details.html">BFR Starship project</a></b> to fulfil their requirements for <b>Mars Endeavour</b>.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br /></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglM_fxTpUP1J8aIQef3oCJ3717sZfez2LoFvE-cE6evSXSZdsBA5g8iADyMBBYJnJlDxD2m0WA8UGFdmygpISCBbrIjObacXtmKSbldQHo7IM_WklubzJubVJ8Y9ZEfj2hFfg0q2IEh_8C/s1600/SpaceX-Crew-Dragon-2_ffa0.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="478" data-original-width="830" height="184" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglM_fxTpUP1J8aIQef3oCJ3717sZfez2LoFvE-cE6evSXSZdsBA5g8iADyMBBYJnJlDxD2m0WA8UGFdmygpISCBbrIjObacXtmKSbldQHo7IM_WklubzJubVJ8Y9ZEfj2hFfg0q2IEh_8C/s320/SpaceX-Crew-Dragon-2_ffa0.jpg" title="Astronaunts Bob Behnken and Doug Hurley inside Dragon capsule" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Astronaunts Bob Behnken and Doug Hurley inside Dragon capsule (source space connect)</td></tr>
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<span style="font-family: "footlight mt light" , "serif"; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><br />We have to see now, in the future, what this company is holding for humanity. There is certainly no doubt that this company is achieving great
heights today and will achieve in the future. <o:p></o:p></span></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7IBWERU-9jUQe-KOJkkPCRB4Wx4shwJniOKUfv0EumcLinBw5dlpWaChYMz1sRjQN8bgLUabKXlgMAbJRFdHh3_T6AwDu6Y_9WLrfm-9tBYjJWFWTMOZzYXcmqrP-QnFLMAnXwj1cOUAF/s1600/Crew_Dragon_Interior_%252821119686299%2529.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Space X Dragon: Engineering of the Dragon that conquered human private spaceflight industry!!! the dynamic frequency ratnadeep das choudhury" border="0" data-original-height="703" data-original-width="1600" height="140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7IBWERU-9jUQe-KOJkkPCRB4Wx4shwJniOKUfv0EumcLinBw5dlpWaChYMz1sRjQN8bgLUabKXlgMAbJRFdHh3_T6AwDu6Y_9WLrfm-9tBYjJWFWTMOZzYXcmqrP-QnFLMAnXwj1cOUAF/s320/Crew_Dragon_Interior_%252821119686299%2529.jpg" title="Crew Dragon's interior" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Crew Dragon's interior</td></tr>
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Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-66043154770504658212020-07-26T22:10:00.001+05:302020-07-27T02:53:58.215+05:30The Quantum Superposition Theorem: A Mathematical Approach<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNkMMsvdzTolCjYaaRUJxbqh0-fFeKHYWlUvDwbUlpzso9yA7IMqYRwIo5wXNuNjIdGzlHuc5VSpgw0qgF_JNej5yHtVeJli3DSML1yIpezKkPzqG9XRcNz2bZrzPit6tXBIxlZKEhfoRq/s1600/explosion-5176090_1920.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The Quantum Superposition Theorem: A Mathematical Approach The Dynamic Frequency " border="0" data-original-height="1067" data-original-width="1600" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNkMMsvdzTolCjYaaRUJxbqh0-fFeKHYWlUvDwbUlpzso9yA7IMqYRwIo5wXNuNjIdGzlHuc5VSpgw0qgF_JNej5yHtVeJli3DSML1yIpezKkPzqG9XRcNz2bZrzPit6tXBIxlZKEhfoRq/s640/explosion-5176090_1920.jpg" title="The Quantum Superposition Theorem: A Mathematical Approach Ratnadeep Das Choudhury The Dynamic Frequency " width="640" /></a></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">In my earlier
articles, I have written mostly about the <b>theoretical side</b> of the story of
<b><a href="https://thedynamicfrequency.blogspot.com/2019/07/quantum-superposition-reason-behind.html">Quantum Superposition</a></b>. Quantum superposition is one of the main corner-stone theory which<b> gives the weirdness to Quantum physics</b> and helps us to solve
<b><a href="https://thedynamicfrequency.blogspot.com/2019/07/explaining-quantum-tunnelling.html">critical problems like Quantum tunneling</a></b>.<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">In my last article, I wrote about the <b><a href="https://thedynamicfrequency.blogspot.com/2020/07/orthogonality-orthonormality-theorem-.html">Orthogonality theorem</a></b> which is a necessary
prerequisite to understand the mathematics behind Quantum Superposition. Along
with this, <b>preliminary calculus with knowledge of probability is required to
understand the following text</b> as it may not seem to be that fancy as you see in
documentaries rather if you understand the text then it is more fascinating and
support the statement: “Facts are stranger than Fiction”.<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">So, without much
further delay, let’s dive into it…..<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">To formulate
superposition principle, firstly we have to consider <b>some potential V(x) and
for this potential Schrodinger’s equation has been solved</b>. This yield a number
of wave-functions </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span>(x)<span style="font-size: 13.5pt;"> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">and their <b>corresponding Eigenvalues for energy </b></span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span></b><span style="font-size: 13.5pt;">. </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">Now, we have to combine each individual solution by linear
summation to construct another function </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝚿</span>(x) <span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">and called it as ‘<b>Super-Wave Function</b>’.<o:p></o:p></span></span></div>
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<span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝚿</span>(x) = <span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span>(x) </span><b style="color: red; font-family: "cambria math", serif; font-size: 18px;">(1)</b></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">Here, </span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span> </b><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">are the <b>expansion coefficients</b></span>. <span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">The expansion
coefficients,<b> </b></span></span><b><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> represent the degree to which the full wavefunction possesses the character of the e</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">igenfunction </span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span></b><span style="color: #0b5394;">.
</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Also, by analogy, you can say that the coefficients are the
projections of the wavefunction on to the eigenfunctions of the operator.</span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">Normalization of </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝚿</span>(x)
<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">over the range of </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span>(x) <span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">leads
us to,<o:p></o:p></span></span></div>
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<span style="color: red;"><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> 𝚿</span> </span><span style="color: red; font-family: "cambria math" , serif; font-size: 18px;">𝚿</span><span style="color: red;">*dx = 1<o:p></o:p></span></div>
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<span style="color: red;"><span style="font-size: 24pt;">∫</span><span style="font-size: 22pt;"> (</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> aᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-size: 22pt;">)(</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-size: 20pt;">ⱼ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> a</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-size: 22pt;">) </span></span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="color: red;">= 1 <b>(2)</b></span><span style="color: #0b5394;"><o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Now we can swap
order if the integration and summation in as it possible without loss of generality
in equation 2. So, after swapping we get, <o:p></o:p></span></span></div>
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<span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 20pt;"> 𝚺</span><span style="font-size: 20pt;">ⱼ</span> <span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ </span></span><span style="color: red; font-family: "cambria math" , serif; font-size: 13.5pt;">a</span><span style="color: red; font-size: 13.5pt;">ⱼ</span><span style="color: red;">*<span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> | 𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩ = 1 <b>(3)</b></span></span><span style="background: white; color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><o:p></o:p></span></div>
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<span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"><br /></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Here, after
rearranging, wave functions for i and j has been represented by Dirac notation.
If you know about the <b><a href="https://thedynamicfrequency.blogspot.com/2020/07/orthogonality-orthonormality-theorem-.html">Orthogonality theorem</a></b>, then you must be familiar with
this notation. If you don’t know about it, then I strongly recommend you to
read it first and then proceed forward.<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">So, after the
above steps, we can use Orthogonality theorem to simplify the equation 3. As we
know that, </span><b><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> | 𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*</b><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"><b>⟩</b> </span><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">can be simplified
to </span><b><span style="font-size: 13.5pt;">ẟ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ^</span><span style="font-family: "footlight mt light" , serif; font-size: 14pt;"><sup>j</sup></span></b></span><span style="background-color: transparent; font-size: 14pt; text-align: left;"><span style="color: #0b5394;">.</span> </span><span style="color: #0b5394;"><span style="background-color: transparent; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;">Again from Orthogonality
theorem, we know that </span><b style="background-color: transparent; text-align: left;"><span style="font-size: 13.5pt;">ẟ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ^</span>
<span style="font-family: "footlight mt light" , serif; font-size: 14pt;"><sup>j</sup></span></b><span style="background-color: transparent; font-family: "footlight mt light" , serif; font-size: 13.5pt; text-align: left;"><b> is 1 only when
i=j</b>. For all other conditions, it is 0. Also, </span><span style="background-color: transparent; font-family: "cambria math" , serif; font-size: 13.5pt; text-align: left;">aᵢ</span><span style="background-color: transparent; text-align: left;"> </span><span style="background-color: transparent; font-family: "cambria math" , serif; font-size: 18px; text-align: left;">aᵢ</span><span style="background-color: transparent; text-align: left;">* = </span><span style="background-color: transparent; font-family: "cambria math" , serif; font-size: 18px; text-align: left;">aᵢ</span><span style="background-color: transparent; text-align: left;">².</span></span></div>
</div>
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<br /></div>
<div class="MsoNormal" style="background: white; line-height: 16.8pt; margin-bottom: 0.0001pt; text-align: justify;">
<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">So, now our equation 3 is modified to,<o:p></o:p></span></span></div>
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<br /></div>
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<span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span> <span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span>² = 1 </span><b style="color: red; font-family: "cambria math", serif; font-size: 18px;">(4)</b></div>
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<br /></div>
<div class="MsoNormal" style="background: white; line-height: 16.8pt; text-align: justify; vertical-align: baseline;">
<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Now, we need to
calculate expectation value of </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝚿</span>(x)
<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">for our further
investigation….<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span></span></div>
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<span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩ = </span><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> 𝚿</span>(<b>H</b> </span><span style="color: red; font-family: "cambria math" , serif; font-size: 18px;">𝚿</span><span style="color: red;">*)dx = <span style="font-size: 24pt;">∫</span><span style="font-size: 22pt;">(</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> aᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-size: 22pt;">){</span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> H</span></b><span style="font-size: 22pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-size: 20pt;">ⱼ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> a</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-size: 22pt;">} </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">dx<span style="mso-spacerun: yes;"> </span><b>(5)</b></span><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><br /></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Here, we used
<b>Hamiltonian operator because the general equation of expectation value demands
an operator, and in the case of expectation value of energy the operator is
Hamiltonian (H)</b>.<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
<div class="MsoNormal" style="background: white; line-height: 16.8pt; text-align: justify; vertical-align: baseline;">
<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Also, from basic
quantum physics, we know,<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="color: red;"><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> H(</span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) </span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">=<b style="mso-bidi-font-weight: normal;"> </b>E</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> 𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span> </span><b style="color: red; font-family: "cambria math", serif; font-size: 18px;">(6)</b></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Now, from eq. 5
and 6,<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩ = </span><span style="font-size: 24pt;">∫</span><span style="font-size: 22pt;">{</span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></b><span style="font-size: 22pt;">(</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> aᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-size: 22pt;">)(</span><span style="font-family: "cambria math" , serif; font-size: 20pt;"> 𝚺</span><span style="font-size: 20pt;">ⱼ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="font-size: 13.5pt;">ⱼ</span>
<span style="font-family: "cambria math" , serif; font-size: 13.5pt;">a</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="font-size: 22pt;">)</span><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> </span><span style="font-size: 22pt;">} </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">dx </span></span></div>
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<span style="color: #0b5394;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Again we have to
follow the same procedure as we did to reduce eq. (2) to eq. (3)<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩ = </span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 20pt;"> 𝚺</span><span style="font-size: 20pt;">ⱼ</span> <span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="font-size: 13.5pt;">ⱼ</span> <span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span> </span><span style="color: red; font-family: "cambria math" , serif; font-size: 18px;">aᵢ</span><span style="color: red;">*<span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> | 𝜓</span><span style="font-size: 13.5pt;">ⱼ</span>*<span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩</span></span><span style="background: white; color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background: white; line-height: 16.8pt; text-align: justify; vertical-align: baseline;">
<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">And by Orthogonality
theorem,<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩ = </span><span style="font-family: "cambria math" , serif; font-size: 20pt;">𝚺</span><span style="font-family: "cambria math" , serif; font-size: 20pt;">ᵢ</span><span style="font-size: 20pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">aᵢ</span>² <b> (7)</b></span><span style="background: white; color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><o:p></o:p></span></div>
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<span style="color: #0b5394;"><br /></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">So,<b style="mso-bidi-font-weight: normal;"> </b>from equation 7, we understand that
<b>the expectation value of energy for the superposition state is just the weighted
sum of the energies of the individual energies and multiplied with the square
of the expansion coefficients (weighted factors)</b>.<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">There are<b> two
major postulates of the quantum mechanics</b> and <b>on the basis of these postulates
we can constitute the superposition theorem</b>. They are as follows:<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><b><span style="color: lime;">1)</span></b><span style="color: #0b5394;"> If a system can
be in any of the individual eigenstates </span></span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">,</span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">then
it can also be in the superposition state as described by equation 1. Thus,<b>
the system need not to be in an eigenstate and may vary</b>. <o:p></o:p></span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><br /></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: lime;"><b>2)</b></span><span style="color: #0b5394;"> When someone proceeds <b>to make a measurement to determine the
state of the system</b>, then the <b>super-wave function </b>or<b> superposition state</b> (</span></span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">𝚿</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">) collapses into one of the eigenstates </span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">. To find the<b> probability
of the system</b> in state </span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;">ᵢ</span>
<span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">can be <b>calculated by</b></span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;"><b> </b></span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt; line-height: 107%;"><span style="color: #0b5394;"><b>using</b></span></span><span style="color: #0b5394; font-family: "cambria math" , serif; font-size: 13.5pt;"><b> aᵢ</b></span><span style="color: #0b5394;"><b>²</b>. </span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">That’s
why you can see <b>equation 4 has value 1</b> as the <b>summation of all probabilities is
always 1</b>.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">In my further articles, you will be introduced to the <b>formulation
of the mathematical structure of the advance theories like perturbation</b> and <b>variation
methods</b>. These theories are the<b> basis of the approximation methods</b> which are
used in <b>quantum mechanics</b>. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><br /></span></span></div>
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Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-15091365476230826772020-07-03T23:55:00.000+05:302020-07-26T16:51:23.992+05:30The Orthogonality Theorem: Mathematical Corner-stone for Superposition theorem and Perturbation Theorem!!!<div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik2G1rq7il3RjI40u7967igo-zxyst80FWaU6EcPDwzDp0Zc1vdsTMjTjg77lZJT2C3evA1IiogEtxH2-RR83dZOfWuG1X4oPjsN0jnTbUitd9LD8NaSm78oezVpwbopTK1adN__c7kgUI/s1600/unnamed.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="630" data-original-width="1200" height="336" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik2G1rq7il3RjI40u7967igo-zxyst80FWaU6EcPDwzDp0Zc1vdsTMjTjg77lZJT2C3evA1IiogEtxH2-RR83dZOfWuG1X4oPjsN0jnTbUitd9LD8NaSm78oezVpwbopTK1adN__c7kgUI/s640/unnamed.jpg" width="640" /></a></div>
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<span style="background: rgb(252 , 252 , 252); font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">Today’s the article will be a little bit more mathematical as this article will deal with
the </span><b>mathematical architecture</b><span style="font-weight: normal;"> and the building blocks of the theories like
</span><b><a href="https://thedynamicfrequency.blogspot.com/2019/07/quantum-superposition-reason-behind.html">Superposition theorem</a> </b>and<b> Perturbation Theorem</b><span style="font-weight: normal;">.<o:p></o:p></span></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">So, without any
further delay, let’s dive in…<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">As always we will
start by considerations as we all know that physics is full of that!!!<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">So, consider there
are two wave functions </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-size: 13.5pt;">𝝍ₙ</span></b> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">and</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b> </b><b style="font-size: medium; text-align: center;"><span style="font-size: 13.5pt;"><span style="font-family: "times new roman";">𝝍ₖ</span></span></b></span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">. Both satisfy the Schrodinger’s equation for some<b>
potential V(x)</b>.<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">Now, if their energies are </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">E</span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="font-family: "times new roman"; font-size: small; text-align: center;"><span style="font-size: 13.5pt;">ₙ</span></span> and
E</span></span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-family: "cambria math" , serif; font-size: small; text-align: center;"><span style="font-size: 13.5pt;"><span style="font-family: "times new roman";">ₖ</span></span></span> respectively then
Orthogonality theorem states that</span><o:p></o:p></span></div>
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<b><span style="color: red;"><span style="font-size: 24pt;">∫</span><span style="font-size: 13.5pt;"> 𝝍ₖ</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*(x) </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red; font-size: 13.5pt;">ₙ</span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">(x) dx =0 (</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">≠ E</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-size: medium;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></b><b><span style="color: red;">) </span>
(1) </b><b style="color: black;"> </b> </span><span style="color: black; font-family: "footlight mt light" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Here,<b> the limits
of the integral is the limit of the system</b> and </span></span><b style="text-align: center;"><span style="color: red;"><span style="color: #0b5394; font-size: 13.5pt;">𝝍</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="color: #0b5394; font-size: 13.5pt;">ₖ</span></b><b style="color: #0b5394;">*</b> </span><span style="color: #0b5394; font-family: "footlight mt light" , serif; font-size: 13.5pt;">is <b>the imaginary part of </b></span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="color: #0b5394; font-size: 13.5pt;">𝝍</span></b><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="color: red;"><span style="color: #0b5394; font-size: 13.5pt;">ₖ</span></span></b><span style="color: #0b5394;">.</span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Well, that’s it…
its Orthogonality theorem’s main statement. But we are <b>here to derive it
</b>also…so let’s finish this task….<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">As I said earlier,
the above-mentioned wave functions obey the Schrodinger’s equations so, </span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/2m)(d</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) + V(x) </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> = </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="color: red;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> <span style="color: red;">(2)</span></span></b></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">And,</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/2m)(d</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) + V(x) </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> = </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="color: red;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> <span style="color: red;">(3)</span></span></b></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Now, if we
multiply </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="font-family: "times new roman"; font-size: small; text-align: center;"><span style="font-size: 13.5pt;">𝝍</span></span><b style="font-size: medium; text-align: center;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="font-family: "times new roman";">ₖ</span></span></span></b>* </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">and</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> <span style="font-family: "times new roman"; font-size: small; text-align: center;"><span style="font-size: 13.5pt;">𝝍</span></span><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-size: 13.5pt;">ₙ</span></b> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">with</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">eq. (2) and the complex conjugate of eq.(3) and then separate
V(x) by subtracting eq.(3) from eq.(2). After all these jumble operations, we
get this expression…</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/2m)[</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> (d</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) - </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">(</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> d</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) ] = (</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> - </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">)</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"> </span> <span style="color: red;">(4)</span></b></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Now, just
integrate this beautiful expression. The limits should be relevant to the
problem or situation. After integrating we get….</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/2m) </span><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> [</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> (d²</span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span><span style="font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span><span style="font-size: 13.5pt;">) - </span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif;"><span style="font-size: 13.5pt;">(</span></span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> d</span><span style="font-family: "cambria math" , serif; font-size: 15px;">²</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx</span><span style="font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span><span style="font-size: 13.5pt;">) ] dx= (</span></span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> - </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) </span><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b> <span style="color: red;">(5)</span></b></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Well, we are
almost there and just a few steps away from deriving the ‘pillar’ condition for
the theorem!!!<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Now, just take the
left side of Eq. (5)…<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">By using some
basic calculus we can write the left side of eq.(5) as …</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span></span></b><b style="background-color: transparent;"><span style="color: red;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/2m) </span><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> (d/dx)[</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> (d</span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx) - </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">(</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">d</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-size: medium;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;">/dx) ] dx</span> <span style="color: red;">
(6)</span></b></span></div>
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<br /></div>
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<br /></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Thus eq.(6)
transforms into…</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">- (ħ</span><span style="background: white; font-family: "cambria math" , serif;"><span style="font-size: 15px;">²</span><span style="font-size: 13.5pt;">/2m) )[</span></span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"> (d</span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">/dx) -</span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">(</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"> d</span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><span style="color: red;">/dx) ]</span><span style="color: black;"> </span><span style="color: red;">(7)</span></span></b><span style="background: white; color: #333333; font-family: "footlight mt light" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">Now, if we make
correct assumption then the <b>wave functions vanish asymptotically</b> and eq.(7)
will be equal to 0. This will mainly occur in boundaries. <o:p></o:p></span></span></div>
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<br /></div>
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<span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">After all these
mathematical drama our lovely eq.(5)
finally converts into this expression….</span><span style="color: #333333;"><o:p></o:p></span></span></div>
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<b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">(</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> - </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">) </span><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium;"><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"> = 0 </span>
<span style="color: red;">(8)</span></b><o:p></o:p></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;"> If </span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></b><b style="text-align: center;"><span style="font-size: 13.5pt;">ₙ</span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">≠</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></b><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><b style="font-family: "Times New Roman"; font-size: medium; text-align: center;"><span style="font-size: 13.5pt;">ₖ</span></b>,</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">then we finally proved
eq.(1).<o:p></o:p></span></span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">So, from eq.(8) we can say
that, <b>any set of functions </b></span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">𝜓</span><span style="font-size: 13.5pt;">ⱼ </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">(x)</span><span style="font-size: 13.5pt;"> </span></b><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><b>such that any two numbers
of the set obey an integral constraint of this form is said to constitute an orthogonal set of functions</b>. <o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b>If each individual member of the set is normalised</b>, then they are
called <b>an orthonormal set of functions</b> and the expression for Orthogonality in
this case can be expressed by <b>Dirac Notation</b> as follows:</span><o:p></o:p></span></div>
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<b><span style="color: red;"><span style="font-size: 24pt;">∫</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> = </span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟨</span><span style="background: white; font-family: "arial" , sans-serif; font-size: 13.5pt;"> </span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">*</span><span style="background: white; font-family: "arial" , sans-serif; font-size: 13.5pt;"> |</span> </span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">𝝍</span></span></b><b><span style="color: red;"><span style="color: red; font-size: 13.5pt;">ₙ</span></span></b><b><span style="color: red;"><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="background: white; font-family: "cambria math" , serif; font-size: 13.5pt;">⟩</span> = <span style="font-size: 13.5pt;">ẟ</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">ⁿ</span></span><span style="color: red;"> </span> </b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"><b><span style="color: black;"> </span><span style="color: red;">(9)</span></b><o:p></o:p></span></div>
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<span style="color: #0b5394;"><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">Here,<b> </b></span><b><span style="font-size: 13.5pt;">ẟ</span></b><b style="font-family: "cambria math", serif; text-align: center;"><span style="font-size: 13.5pt;"><span style="font-family: "times new roman";">ₖ</span></span></b><b><span style="font-size: 13.5pt;">ⁿ</span></b> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">represents
the <b>Kronecker Delta</b> and defined by:</span><o:p></o:p></span></div>
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<b><span style="color: red;"><span style="font-size: 13.5pt;">ẟ</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">ⁿ</span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"> = 1<span style="mso-spacerun: yes;"> </span>(k=n)<o:p></o:p></span></span></b></div>
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<b><span style="color: red;"><span style="font-size: 13.5pt;">ẟ</span></span></b><b style="font-family: "cambria math", serif;"><span style="color: red;"><span style="font-size: 13.5pt;"><span style="color: red; font-family: "times new roman";">ₖ</span></span></span></b><b><span style="color: red;"><span style="font-size: 13.5pt;">ⁿ</span> </span></b><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><b><span style="color: red;">= 0 (otherwise)</span></b><o:p></o:p></span></div>
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<span style="color: #0b5394;"><span style="background: white; font-family: "footlight mt light" , serif; font-size: 13.5pt;">Orthogonality
helps us to answers an important question, i.e. <b>why two wave functions don’t overlap
each other? </b>But, now with the help of the above derivation, we found out that
only and only <b>if parameter k and n are same then only the integral of eq.(9)
will have some value</b>. Otherwise, it will be<b> zero</b>. Also from eq.(8), we can say
that<b> </b></span><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></b><b style="text-align: center;"><span style="font-size: 13.5pt;">ₙ</span></b><b><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">≠</span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;"> </span><span style="font-family: "cambria math" , serif; font-size: 13.5pt;">E</span></b><span style="font-family: "times new roman" , serif; font-size: 13.5pt;"><b style="font-family: "cambria math", serif; font-size: medium; text-align: center;"><span style="font-size: 13.5pt;"><span style="font-family: "times new roman";">ₖ</span></span></b> </span><span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;">and it proves our statement.<o:p></o:p></span></span></div>
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<span style="font-family: "footlight mt light" , serif; font-size: 13.5pt;"><span style="color: #0b5394;">So, that’s it…that’s all
you need to know about Orthogonality and Orthonormality to start understanding
the mathematical structure of Quantum Superposition and Perturbation theorem.
<b>Till now, on this website I only explored the theoretical side of the
Superposition, but now onwards I will try to explain the mathematical architecture
of it</b>. So, be with me and we will explore this together.</span><o:p></o:p></span></div>
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Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0tag:blogger.com,1999:blog-3578987301814134971.post-76717200627539942142020-06-09T01:09:00.000+05:302020-06-09T01:09:55.298+05:30Einstein’s Greatest Mistakes: 1st Anniversary Special Post!!!<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">Well…<b>Einstein</b>…we
all know about him!!!<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">His contribution
in physics was so great and influential that it is <b>directly responsible</b> for <b>the
creation of a completely new field in physics</b>. He helped us to solve problems
like mercury’s orbit and his <b><a href="https://thedynamicfrequency.blogspot.com/search/label/Relativity" target="_blank">theory of relativity</a></b> helped us to understand <b><a href="https://thedynamicfrequency.blogspot.com/2019/09/formation-of-black-holes.html" target="_blank">black holes</a></b>. For his theory of relativity, today we all are enjoying various services
from the satellites. Well, if that’s not enough….his joint paper with <b>Satyandra
Nath Bose</b>, the <b>Bose-Einstein Statistics</b> directly helped in the creation of
today’s <b><a href="https://thedynamicfrequency.blogspot.com/search/label/Quantum%20Mechanics" target="_blank">Quantum Mechanics</a></b>. And consequently, he became the <b>Time's Person of the
Century for the 20<sup>th</sup> century</b>.<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">It’s hard to
imagine that such a person made some serious mistakes in his field but, as they
said “<b>Facts are Stranger than Fiction</b>” and this case is not an exception. So,
are you ready to dig into this???<o:p></o:p></span></span></div>
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<span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><b><span style="color: magenta;">Well, before
starting, I want to thank everybody who consistently visited my blog and gave
their valuable suggestion to me to improve this blog. For the last 1 year, I am
just trying to enrich the knowledge of people without any cost. The motto,
target and enthusiasm is still the same as it was in day 1 and it will remain the same for
coming years!!!</span></b><span style="color: #0b5394;"><o:p></o:p></span></span></div>
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<span style="background: white; font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;">So, let’s dig into
this article….<o:p></o:p></span></span></div>
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<b><span style="color: red; font-size: x-large;"><span style="background: white; font-family: "Footlight MT Light", serif;">1) Adding Cosmological
constant (</span><span style="background: rgb(252, 252, 252); font-family: Cambria, serif;">Λ</span></span></b><strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;">):</span><span style="color: #0b5394; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGByxEo3JF63xQU0-tBfsCpgOPvjLV-9_rbBJyMdiNTvK_pv4NI5D_fGJV3-Dl7TXliw27qLjSdL6_gBgkyzswYrocEwHDjYecmZkHQX-paltR63o50ngodOYCC0ZcLE21faqN-kWTdX4/s1600/800px-Lambda-Cold_Dark_Matter%252C_Accelerated_Expansion_of_the_Universe%252C_Big_Bang-Inflation.jpg" imageanchor="1"><img border="0" data-original-height="451" data-original-width="800" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbGByxEo3JF63xQU0-tBfsCpgOPvjLV-9_rbBJyMdiNTvK_pv4NI5D_fGJV3-Dl7TXliw27qLjSdL6_gBgkyzswYrocEwHDjYecmZkHQX-paltR63o50ngodOYCC0ZcLE21faqN-kWTdX4/s320/800px-Lambda-Cold_Dark_Matter%252C_Accelerated_Expansion_of_the_Universe%252C_Big_Bang-Inflation.jpg" width="320" /></a></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt; font-weight: normal;"><o:p><span style="color: #0b5394;"><br /></span></o:p></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">This was a
great mistake. The gravity of this was so much great that the Einstein himself referred this as </span>his greatest mistake<span style="font-weight: normal;">. Initially, </span>Einstein thought that the universe was
static<span style="font-weight: normal;"> and </span>unchanging<span style="font-weight: normal;">. But, his equations constantly showing that it wasn’t. To
rectify this error (which is actually correct), </span>Einstein added cosmological constant<span style="font-weight: normal;">
in his equation of relativity. This rectification was </span>unstable anyways<span style="font-weight: normal;"> as a
</span>slightly denser region than the normal would collapse and slightly less dense
region would expand. That means everything should be perfect<span style="font-weight: normal;">.</span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">If Einstein didn’t
add that constant then he would predict the expansion of the universe before
</span>Friedman and Lamaitre and before Hubble found the evidence of it<span style="font-weight: normal;">.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">Later,
Einstein rectify his mistake and </span>removed the cosmological constant <span style="font-weight: normal;">from the
equation.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;">2) Accepting
Gravitational Waves:</span><span style="color: #0b5394; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi45X6-6QIm_1OIt1FoJgX5oUoZRHR5x5jLYyXNNpoIB3caEpGcAYEck-nlNy4S4f01rd4vkjfrN_5G3PchSFSvurnpdiHZ_0UZg5EaSTvOIzSFL3DP7_kNP_AaKQ7XpX9j-o5wN5H7hAWg/s1600/Wavy.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="200" data-original-width="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi45X6-6QIm_1OIt1FoJgX5oUoZRHR5x5jLYyXNNpoIB3caEpGcAYEck-nlNy4S4f01rd4vkjfrN_5G3PchSFSvurnpdiHZ_0UZg5EaSTvOIzSFL3DP7_kNP_AaKQ7XpX9j-o5wN5H7hAWg/s1600/Wavy.gif" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Illustration of Gravitational Waves</td></tr>
</tbody></table>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">Although, Einstein
</span>predicted the gravitational waves<span style="font-weight: normal;"> in </span>1916<span style="font-weight: normal;"> but he believed that they would be
incredibly </span>hard to measure<span style="font-weight: normal;"> and </span>also doubted the existence of the black holes
<span style="font-weight: normal;">that may be causing them.</span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">But, scientists
of </span>LIGO<span style="font-weight: normal;"> detected gravitational waves successfully for the </span>first time<span style="font-weight: normal;"> in </span>2015<span style="font-weight: normal;">,
about </span>100 years after Einstein’s prediction<span style="font-weight: normal;">.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;">3) Rejecting
Quantum Mechanical model:</span><span style="color: #0b5394; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqa72V0WqmyL41-GelIC3RJo6jWivN1YTUiZ0Q03XjINizd4nDON0wk5qML-QJhUKRbiDmgmDaVBysx0s8r-jj6kCdG9CdDLh6yS3HFnSPCyo7_WJkzmLL9NCM117uQ0Uk7dQpTCPraJQx/s1600/cassini_orbits.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="1280" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqa72V0WqmyL41-GelIC3RJo6jWivN1YTUiZ0Q03XjINizd4nDON0wk5qML-QJhUKRbiDmgmDaVBysx0s8r-jj6kCdG9CdDLh6yS3HFnSPCyo7_WJkzmLL9NCM117uQ0Uk7dQpTCPraJQx/s320/cassini_orbits.gif" width="320" /></a></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">Einstein
contributed significantly in the field of quantum mechanics by developing theories
like the </span><a href="https://thedynamicfrequency.blogspot.com/2019/06/a-little-explanation-of-photoelectric.html" target="_blank">photoelectric effect</a><span style="font-weight: normal;"> and developing mathematical structure like
Bose-Einstein statistics. </span>Even Einstein won Nobel Prize for photoelectric
effect<span style="font-weight: normal;">.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">But Einstein didn’t
liked the concepts of the quantum world. He called the ‘</span><a href="https://thedynamicfrequency.blogspot.com/search/label/Quantum%20Entanglement" target="_blank">Quantum Entanglement</a><span style="font-weight: normal;">’ as ‘</span>Spooky
Force<span style="font-weight: normal;">’. Moreover, he also rejected the idea of </span><a href="https://thedynamicfrequency.blogspot.com/2019/06/heisenberg-uncertainty-principle-simplified.html" target="_blank">Heisenberg’s Uncertainty principle</a><span style="font-weight: normal;">. He believed in a </span>deterministic nature<span style="font-weight: normal;"> where everything can be pre-calculated,
which was not the case with the</span> <a href="https://thedynamicfrequency.blogspot.com/search/label/Quantum" target="_blank">Quantum world</a><span style="font-weight: normal;">.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;">4) Einstein’s
wrong approach to find Unification theory:</span><span style="color: #0b5394; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><b>Unification
theory</b> or <b>Grand Unification Theory</b><span style="font-weight: normal;"> is theory by which </span><b>all of the forces and phenomena
of nature can be explained by using few simple rules, equations and parameters</b><span style="font-weight: normal;">.
<o:p></o:p></span></span></span></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">For example,
</span>Maxwell’s law<span style="font-weight: normal;"> unifies both </span>Electric<span style="font-weight: normal;"> and </span>magnetic forces<span style="font-weight: normal;"> and forms
</span>Electromagnetism<span style="font-weight: normal;">. <o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;"><br /></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt; font-weight: normal;"><span style="color: #0b5394;">Ever-since, we
all are trying to unify all those forces and Einstein was no exception.<o:p></o:p></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">In the </span>last
years of his life<span style="font-weight: normal;">, he tried to </span>unify electromagnetism and gravitation<span style="font-weight: normal;">. Many
experiments revealed various new rules but, Einstein ignored all those. Even it
was also showed that the</span> strong nuclear force<span style="font-weight: normal;"> can be</span> unified with
Electromagnetism and weak force<span style="font-weight: normal;">. This unification can be seen in the current </span><a href="https://thedynamicfrequency.blogspot.com/search/label/Standard%20Model" target="_blank">Standard Model</a><span style="font-weight: normal;">. But, Einstein </span>decided to stick with gravity and electromagnetism<span style="font-weight: normal;"> and try
to unify those without using new rules which results in failure. <o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">Today, the scenario
changed. </span>Grand unified theory now can be explained by various theories like
String theory. But, still, those are yet to be proven<span style="font-weight: normal;">.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif;"><span style="color: red; font-size: x-large;">5) Einstein
simple mistake in the derivation of E=mc^2:</span><span style="color: #0b5394; font-size: 13.5pt; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt; font-weight: normal;"><o:p><span style="color: #0b5394;"><br /></span></o:p></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">After reading
this heading, </span>you probably thinking that I have lost my mind<span style="font-weight: normal;">. But believe me, </span>I
haven’t <span style="font-weight: normal;">…..<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">It is
absolutely </span>correct that Einstein derived E= mc^2<span style="font-weight: normal;">. But, the strange truth is
that he </span>derived it for only rest particle only<span style="font-weight: normal;">. In other words, you can say
that his </span>derivation meant for frame-dependent observation only<span style="font-weight: normal;">. It is hard to
believe that he did this even after</span> setting the laws for relativistic phenomena
in his Special Theory of Relativity paper<span style="font-weight: normal;">. </span><span style="font-weight: normal;"> <o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">It wasn’t six
years later, when </span>Nobel laureate Max Von Laue<span style="font-weight: normal;"> showed that one must </span>remove the
idea of kinetic energy to fix this error<span style="font-weight: normal;"> and by this way ultimately fixing the
error.<o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: #0b5394;"><span style="font-weight: normal;">It is hard to
believe that even geniuses like Einstein </span>made some critical mistakes<span style="font-weight: normal;"> but, at
the end </span>we should remember that they are also humans and humans make mistake<span style="font-weight: normal;">s. <o:p></o:p></span></span></span></strong></div>
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<strong><span style="background: rgb(252, 252, 252); font-family: "Footlight MT Light", serif; font-size: 13.5pt;"><span style="color: magenta;">At last, I
again want to thank all of you who stayed and supported me for last one
year. Thank you very much!!!</span><span style="color: #333333; font-weight: normal;"><o:p></o:p></span></span></strong></div>
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<span style="background-position: 0px 0px; border: 0px; font-family: "footlight mt light" , serif; font-size: 13.5pt; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background-position: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;"><b style="background-position: 0px 0px; border: 0px; color: #474747; outline: 0px; padding: 0px; vertical-align: baseline;">To know basics of the quantum world, astronomy and space exploration you can check out my book <a href="https://www.amazon.in/dp/B07TBSP6JT" style="background-position: 0px 0px; border: 0px; color: #e94bc3; outline: 0px; padding: 0px; transition: all 0.17s ease 0s; vertical-align: baseline;" target="_blank">"Through the wormhole" on amazon kindle</a>.</b></span></span></div>
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<em style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; box-sizing: border-box; font-weight: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background-position: 0px 0px; border: 1pt none; font-family: "footlight mt light" , serif; font-size: 13.5pt; font-style: normal; outline: 0px; padding: 0cm; vertical-align: baseline;"><span style="background-position: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;">Thanks for reading.....</span></span></em></div>
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<em style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; box-sizing: border-box; font-weight: inherit; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background-position: 0px 0px; border: 1pt none; font-family: "footlight mt light" , serif; font-size: 13.5pt; font-style: normal; outline: 0px; padding: 0cm; vertical-align: baseline;"><span style="background-position: 0px 0px; border: 0px; color: #0b5394; outline: 0px; padding: 0px; vertical-align: baseline;">See you next time!!!</span></span></em></div>
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<span style="background-position: 0px 0px; border: 0px; color: lime; font-family: "footlight mt light" , serif; outline: 0px; padding: 0px; vertical-align: baseline;"><span style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; font-size: 18px; outline: 0px; padding: 0px; vertical-align: baseline;"> -Ratnadeep Das Choudhury</span></span></div>
<div style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: 0px 0px; background-repeat: initial; background-size: initial; border: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">
<span style="background-position: 0px 0px; border: 0px; color: lime; font-family: "footlight mt light" , serif; font-size: 18px; outline: 0px; padding: 0px; vertical-align: baseline;"> Founder and Writer of The Dynamic Frequency</span></div>
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Ratnadeep Das Choudhuryhttp://www.blogger.com/profile/09215954499894048488noreply@blogger.com0