Understanding Schrödinger’s Time-Dependent Equation and need of it!!!

In my previous article about Schrödinger’s equation, I thoroughly derive Schrödinger’s Time-Independent equation, but still now we are not independent to understand our quantum world fully!!!

For that we need another and more sensible version of Schrödinger’s wave equation. Any sensible wave equation should be both space and time dependent. In the preceding derivation, time dependence was overlooked by concerning ourselves only with derivatives of y with respect to x. In doing so, any knowledge of the direction sense of the wave pattern was forgone.

But there is no harm to derive and learn the previous derivation as it will behave like a pseudo-derivation of time-dependent SE.

Now, to derive time-dependent SE, we need knowledge of some equations which are as follows:

                                                       λ=h/p                      (de-Broglie’s Wavelength)

                                                E=hv                 (Plank’s Energy-Frequency Relation)

                                                w=2πv                (Definition of Angular Frequency)         

Where, λ= wavelength

             h=plank’s Constant

             p=momentum

             E=Energy

             v=frequency

Now, as we deed in last derivation, we will firstly take the prototype wave-function,

     Ψ=Asin (px/ ħ-wt)                     …………….(1)

                                          

Using Plank’s Energy-Frequency Relation and Definition of Angular Frequency,

                                                

                                               Ψ=Asin [(px-Et)/ ħ]                 ………………(2)

Now, derivative of equation 2 with respect to time,

            ∂Ψ/∂t = (-E/ ħ ) A cos[(px-Et)/ ħ]   …………….(3)

But equation 3 has a serious problem. Here, wave-function is represented by a function. It should include wave-function only, not its function as it should be independent of the form of any of its particular solution.

To rectify this, we will use one identity of basic trigonometry,

sin^2 θ +cos^2 θ=1

Now,                 A^2 sin^2[(px-Et)/ ħ] + A^2 cos^2 [(px-Et)/ ħ] =A^2  ………..(4)

From equation 2 and 4, we get,

A cos [(px-Et)/ ħ]   =±√A^2- Ψ^2      ……………..(5)

From equation 5 and 3,

                                               E= [∓ ħ/√A^2- Ψ^2](∂Ψ/∂t)

Substituting this value to the right side of the time independent equation,

                      - ħ^2/2m(∂^2 Ψ /∂x^2) + V(x)y=[∓ ħ/√A^2- Ψ^2](∂Ψ/∂t)

But this result has many serious problems:

1) Amplitude of the wave should not appear in what is presumed to be a physical    

    law

2) Presence of square root in denominator of the right side makes this differential   

    equation non-linear

3) There is a sign ambiguity

Faced with all this situation Schrödinger got an idea to modify the wave equation like this,

Ψ=A exp[i(px-Et)/ ħ]                 

where, i=√-1

By this assumption, now we can leave time independent SE unchanged.

Now,                                        E=( i ħ/ Ψ)(∂Ψ/∂t)

Substituting this result in time independent equation, we get,

                                  - ħ^2/2m(∂^2 Ψ /∂x^2) + V(x) Ψ =( i ħ)(∂Ψ/∂t)

This equation is called as Schrödinger’s Time-dependent equation.

Okay…okay…finally we got our equation....So,can we now take rest….????

Hmm...actually the answer is NO!!!! 
Imagine....what will happen if I say in-spite of getting our equation, still there is a major mistake lying in this equation…..

Okay...for that you have to wait for my next blog.....my next blog will be on that mistake!!!

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Thanks for reading.....

See you next time!!!

                                                                      -Ratnadeep Das Choudhury

                                                      Founder and Writer of The Dynamic Frequency