Derivations

# 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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*See you next time!!!*

-Ratnadeep Das Choudhury

Founder and Writer of The Dynamic Frequency

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