Shrodinger Equation, where did it come from?

[Nick89]

Hi,

I was just wondering this lately: where does the Shrodinger Equation come from? How was it 'invented'?
Our teacher told us that it cannot be derived from any classical mechanics (quite obviously) so how did Shrodinger come up with it?

I can hardly believe he just postulated that a wavefunction would obbey that equation, and it "happened" to obbey measurements..?

I can't find this anywhere, not even in my QM textbook (by Griffiths). Actually the first chapter begins by stating the SE... No explanation whatsoever!

 

[Dadface]

I am not sure if my history is correct here but I believe that his equation was borrowed from classical theory and he found that it could be adapted to encompass De Broglies idea of the wave nature of matter.

 

[jtbell]

how did Shrodinger come up with it?

https://www.physicsforums.com/showpost.php?p=418069&postcount=5

 

[Fredrik]

I don't know exactly, but I think it went something like this:

de Broglie's work suggested that matter had wave properties. So let's try writing down a function describing a wave. $exp(-iEt+i\vec p\cdot \vec x)$ is a plane wave propagating in the direction of $\vec p$. That suggests that $\vec p$ is proportonal to the velocity. It could be the velocity itself, or the momentum. If this exponential is the solution of a differential equation, then this equation is also going to tell us the relationship between E and $\vec p$. What equation will give us the relativistic relationship between energy and momentum, $E^2=\vec p^2+m^2$? The answer is the Klein-Gordon equation. That didn't work out very well, so let's try the equation that gives us the non-relativistic relationship between energy and momentum, $E=\vec p^2/2m$. The result is the Schrödinger equation.

I'm not sure about the details, but I'm pretty sure that I've read that the solutions were found before the equation, and that the Klein-Gordon equation was found before the Schrödinger equation. (By the way, I'm using units such that $\hbar=1$).

Edit: My speculations can safely be ignored. Jtbell seems to have actually read the original papers. I consider that a form of cheating.

 

[Nick89]

Thanks, that actually makes sense :)

Our teacher did give us a quick "semi-derivation" simply by explaining what the derivatives for example physically represent and that it is at least probable that it is correct, but this is much more satisfying :)