In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles.(adsbygoogle = window.adsbygoogle || []).push({});

Unfortunately I don't think page 262 is showing so I'll explain the gist of it:

"In his initial paper" he considers the reduced Hamilton-Jacobi equation

[itex]\frac{1}{2m}[(\frac{\partial S}{\partial x})^2 \ + \ (\frac{\partial S}{\partial y})^2 \ + \ (\frac{\partial S}{\partial z})^2] \ + \ V(x,y,z) \ - \ E \ = \ 0[/itex]

for a single particle of mass [itex]m[/itex] in an arbitrary force field described by a potential [itex]V = V(x,y,z)[/itex]

& with a change of variables [itex]S \ = \ K\log(\Psi)[/itex], (where [itex]K[/itex] will turn out to be [itex] \frac{h}{2\pi}[/itex]) it reduces to

[itex] \frac{K^2}{2m}[(\frac{\partial \Psi}{\partial x})^2 \ + \ (\frac{\partial \Psi}{\partial y})^2 \ + \ (\frac{\partial \Psi}{\partial z})^2] \ + \ (V \ - \ E)\Psi^2 \ = 0[/itex].

Now instead of solving this he, randomly from my point of view, choosed to integrate over space

[itex]I \ = \ \int\int\int_\mathcal{V}(\frac{K^2}{2m}[(\frac{\partial \Psi}{\partial x})^2 \ + \ (\frac{\partial \Psi}{\partial y})^2 \ + \ (\frac{\partial \Psi}{\partial z})^2] \ + \ (V \ - \ E)\Psi^2)dxdydz[/itex]

& then extremizes this integral which gives us the Schrodinger equation.

Apparently as the book then claims on page 264 it is only after this derivation that he sought to connect his idea's to deBroglie's wave-particle duality.

Thus I have three questions,

1) What is the justification for Feynman's famous quote:

"Where did we get that [Schrödinger's equation] from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger"

The Feynman Lectures on Physics

in light of the above derivation. I note that all the derivations I've seen of the Schrodinger equation doing something like using operators such as [itex]i\frac{h}{2\pi}\frac{\partial}{\partial t} \ = \ E[/itex] to derive it always mention it's merely heuristic, yet what Schrodinger apparently originally did seems like a roundabout way of solving the Hamilton-Jacobi equation with no heuristic-ness in sight. What subtleties am I missing here? Why would I be a fool to arrogantly 'correct' someone who says Schrodinger is not derivable from anything you know?

2) Is the mathematical trick Schrodinger has used something you can use to solve problems?

3) Why can't you use this exact derivation in the relativistic case?

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# Variational Derivation of Schrodinger Equation

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