Mistakes of Schrödinger time dependent equation and their rectifications!!!!

In my previous article, I derived Schrödinger’s Time-dependent equation in a very detailed manner. If you haven’t seen it…I request you to please check it out first and then read this article to better understand the concept….

At the end of that article, I told that despite our success in arriving at a linear time-dependent differential equation, a major final problem remains.

       - ħ^2/2m(∂^2 Ψ /∂x^2) + V(x) Ψ =( i ħ)(∂Ψ/∂t)     {Time-Dependent Eq.}

In the process to derive the equation, we completely ignored the fact that this final equation is neither a wave to the left or to the right.

To rectify it, we have to use the way of old mathematicians…….

          sin θ =(1/2i)(e^iθ- e^-iθ)         {Euler’s identity}

Now the right-travelling matter wave can be written as,

                                    Ψ=A sin[i(px-Et)/ ħ]

                                   =[A/2i][e^{(i/ ħ)(px-Et)} – e^{(-i/ ħ)(px-Et)}]

                

Now, surprisingly a purely right or left moving wave shows us an unacceptable time dependent wave equation. Also, either of the individual exponential components doesn’t lead to a sensible differential equation.

So, to debug this problem Schrödinger proposed adopting one of the exponential term as right directed or +x direction wave and another as left directed or –x direction wave.

So,                     Ψ (right) (x, y, z) = A e^ {(i/ ħ) (px-Et)}

And,                  Ψ (left) (x, y, z) = A e^ {(-i/ ħ) (px-Et)}

Where, the factor of 1/2i is absorbed in amplitude (A).

In one of my earlier post I showed the derivation of Schrödinger’s time independent equation. If you compared that equation with this then we find that,

 Ψ (x, y, z) = ψ(x) e^ -iwt

Where, ψ(x) represents the solution of time-independent equation and w=E/ ħ.

That’s it….Easy right???......

Now, we are ready to go into the quantum-verse mathematically…..In my subsequent articles, I will take you deep inside the quantum-verse mathematically and explain each phenomenon in most simple way……

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Thanks for reading.....

See you next time!!!

                                                                    -Ratnadeep Das Choudhury

                                                      Founder and Writer of The Dynamic Frequency