I read in volume III of Landau's and Lifshitz's Theoretical Physics that the action of classical mechanics can be seen as the complex phase of the wave function of quantum mechanics. I also read a reference somewhere else, that it was in a 1948 paper1 that Richard Feynman demonstrated this relationship. Landau and Lifshitz were not very informative, and I had no access to Feynman's old paper (I've since obtained a copy, but it's a Good Thing that I didn't have it earlier, as it gave me the motivation to do these calculations on my own), so I decided to work this out myself, at least in the simple case of a point particle in a potential field.

The Schrödinger equation of a point particle is well known:

\[i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial q^2}+V\psi.\]

What if ψ is expressed as ρeiS/ħ, where S is an arbitrary function of time, the coordinates, and perhaps momenta? Let's do the substitution, dropping the eiS/ħ part as it appears as a factor on both sides of the equation:

\[i\hbar\frac{\partial\rho}{\partial t}-\rho\frac{\partial S}{\partial t}=-\frac{\hbar^2}{2m}\left[\frac{\partial^2\rho}{\partial q^2}+\frac{2i}{\hbar}\frac{\partial\rho}{\partial q}\frac{\partial S}{\partial q}+\frac{i\rho}{\hbar}\frac{\partial^2S}{\partial q^2}-\frac{\rho}{\hbar^2}\left(\frac{\partial S}{\partial q}\right)^2\right]+V\rho.\]

Separating the real and imaginary parts yields two equations. The equation for the real part looks like this:

\[-\rho\frac{\partial S}{\partial t}=-\frac{\hbar^2}{2m}\left[\frac{\partial^2\rho}{\partial q^2}-\frac{\rho}{\hbar^2}\left(\frac{\partial S}{\partial q}\right)^2\right]+V\rho,\]

which, when divided by ρ, can be simplified as

\[-\frac{\partial S}{\partial t}=\frac{1}{2m}\left(\frac{\partial S}{\partial q}\right)^2-\frac{\hbar^2}{2m\rho}\frac{\partial^2\rho}{\partial q^2}+V.\]

The smallness of ħ suggests that, at least for macroscopic systems, the second term on the right hand side can be eliminated, leaving us with the equation

\[-\frac{\partial S}{\partial t}=\frac{1}{2m}\left(\frac{\partial S}{\partial q}\right)^2+V,\]

which is the equation of motion for a point particle governed by the action S.

As a footnote of sorts, I now found that this relationship is also demonstrated in §§31-32 of Dirac's book2.

1R. P. Feynman: "Space-time Approach to Non-relativistic Quantum Mechanics", Rev. Mod. Physics, Vol 20, pp. 367-387, April 1948
2P. A. M. Dirac: "The Principles of Quantum Mechanics", Oxford University Press, 1958