5

in his paper "Quantisierung als Eigenwertproblem" he begins by introducing the Hamilton-Jacobi equation: $$H\left (q,\frac{\partial{S}}{\partial{q}} \right)=E\tag{1}$$ such that $S$ is Hamilton's characteristic function (usually denoted $W$).

Then he performs the Ansatz $$S=K \ln\psi\tag{2}$$ and writes the hamiltonian explicitly to arrive to the following equation:

$$\frac{(\nabla\psi)^2}{2m}-\frac{2m}{K^2}\left (E+\frac{e^2}{r} \right)\psi^2=0.\tag{1"}$$

so far so good, I understand the steps. but then he integrates the equation over all space and regards it as a functional to be minimized, he writes (Equation (3) in the paper):

$$\delta J=\delta \int dxdydz \left[ \frac{(\nabla\psi)^2}{2m}-\frac{2m}{K^2} \left (E+\frac{e^2}{r} \right)\psi^2\right ]=0\tag{3}$$

and by performing the variation "The Standard way" as he writes in German: he gets the well known form for the Schrödinger equation for the Hydrogen Atom. (equations (4) and (5) in the linked paper.)

I understand how to perform the variation, but why did he define such a functional in the first place? why is the volume integral over the eigenvalue problem a functional to be minimized?

Tomka
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    It's a handwaving manoeuvre. You can't derive the Schrodinger equation from classical mechanics, you can only motivate it. Shroedinger knew he needed a wave equation because Debye had said, in response to S trying get Bohr's model from De Broglie waves, that "if you have waves you need a wave equation". This manoevre gets you a wave equation. – mike stone Mar 07 '21 at 15:32
  • That is very disappointing. Surely there has to be some justification or interpretation to that move. Even if S couldn't tell the meaning that doesnt mean there isnt one. Maybe someone has found a physical interpretation to the quantity that is that integral? – Tomka Mar 07 '21 at 16:13
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    @PapaJonathan basic physics laws are inferred from observations or guessed. It is the pavement on which the rest of the theory is built, there is nothing deeper there that we know of. There may be deeper theory behind Schroedinger's equation but so far we do not have it. – Ján Lalinský Mar 07 '21 at 18:20
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    Schroedinger's method is just "heuristics" or playing with related ideas from theoretical mechanics, geometrical optics, seeing what can stick. In his case it worked and produced revolutionary results. But today we do not think there is much need/use for his variational argument: his result , the Schroedinger equation, is a better "pavement" to build the theory on. – Ján Lalinský Mar 07 '21 at 18:22

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