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In Schrödinger's Quantisation as an Eigenvalue Problem he solves the Hydrogen atom through a precursor of Schrödinger's Equation, derived from the Hamilton-Jacobi equation through a variational method as follows. Starting from the Hamilton-Jacobi Equation $$H\left(q,\frac{\partial S}{\partial q}\right)=E$$ We introduce a variable $\psi$ such that $$S=K\ln\psi$$ The H-E Equation becomes $$H\left(q,\frac{K}{\psi}\frac{\partial \psi}{\partial q}\right)=E$$ Which, in the non-relativistic limit, can be expressed in a quadratic form, such as (for the hydrogen atom) $$\left(\frac{\partial \psi}{\partial x}\right)^2+\left(\frac{\partial \psi}{\partial y}\right)^2+\left(\frac{\partial \psi}{\partial z}^2\right)-\frac{2m}{K^2}\left(E+\frac{e^2}{r}\right)\psi^2=0$$ Schrödinger then suggests that, instead of solving this equation, to find an equation $\psi$, such that the integral over all space of the previous equation is stationary, such that $$\delta J=\delta\int\int\int dxdydz\left[\left(\frac{\partial \psi}{\partial x}\right)^2+\left(\frac{\partial \psi}{\partial y}\right)^2+\left(\frac{\partial \psi}{\partial z}^2\right)-\frac{2m}{K^2}\left(E+\frac{e^2}{r}\right)\psi^2\right]=0$$ My first problem is I don't understand the motivation for this. Is there a reason or was it just a lucky guess?

Second, Schrödinger then makes this equation into $$\frac{1}{2}\delta J =\int df \delta\psi \frac {\partial \psi}{\partial n}-\int\int\int dx dy dz d\psi\left[\nabla^2\psi+\frac{2m}{K^2}\left(E+\frac{e^2}{r}\right)\psi\right]=0$$ Where $df$ is the surface element of the infinite closed surface inside of which the integral is evaluated.

I don't know how to get this integral from the first one, and I have no idea what $n$ is. Please help.

Phineas Nicolson
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    See https://physics.stackexchange.com/q/69982/25851 also https://physics.stackexchange.com/q/77171/25851 – bolbteppa Aug 20 '18 at 00:00
  • He uses integration by parts, or the multivariable 'Green identities' if you prefer, to get the last equation. – bolbteppa Aug 20 '18 at 00:02
  • He says in his paper "I am aware this formulation is not entirely unambiguous". It makes absolutely no sense to do this classically, and amounts to averaging, for no (classical) reason, over the region i.e. incorporating probability. The substitution $S = K \ln \Psi$ amounts to the 'quasi-classical approximation' found in Landau's QM section 6 which is set up from first principles in that book by immediately declaring classical mechanics is wrong in the Heisenberg uncertainty principle stating paths don't exist, then building a new theory based on this. – bolbteppa Aug 20 '18 at 00:06
  • This https://www.amazon.com/Probability-Schrodingers-Mechanics-David-Cook/dp/9812381910/ is a book which tries to generalize Schrodinger's method given here and ends up basically saying you still need to do this random averaging thing to get QM if starting from classical mechanics, and amounts to hiding in the math what Landau makes clear from the start - the paths just don't exist... – bolbteppa Aug 20 '18 at 00:10
  • Thanks. So, it is a lucky guess, it seems? I was asking because it seems all other derivations of quantum mechanics have some sort of motivation behind the leap of logic to get the new theory. Schrödinger's derivation of the Wave Equation in An Undulatory Theory of the Mechanics of Atoms and Molecules is motivated by the hamiltonian analogy between Classical Mechanics and Optics. Heisenberg's derivation of Matrix Mechanics is motivated by the Fourier expansion of particle movement and the old quantum condition. It just feels dissapointing that this one case is just a lucky guess. – Phineas Nicolson Aug 20 '18 at 00:53
  • At the beginning of the second paper he calls $S = K \ln \Psi$ "unintelligible", and varying the integral of the equation he gets from it "equally incomprehensible", but then seems to imply (and at the end of the first paper) he's vaguely motivated by DeBroglie and linking waves to particles... – bolbteppa Aug 20 '18 at 02:24
  • In the derivation of the Eikonal equation in the geometric optics approximation in classical field theory (e.g. Landau vol. 2 Ch. 7) one sees how the action behaves similarly to the argument of a wave, so setting $S = K \ln \Psi$ is not out of nowehere if still completely unjustifiable from a classical perspective, but then integrating and then extremizing the equation you get is even more unjustifiable, basically you're 'averaging' it to try to turn a particle equation into a wave equation I guess, destroying the idea of definite paths in the process. – bolbteppa Aug 20 '18 at 02:32

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