Physics Notes: The action and the wave function

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:

iħ 
∂φ
  = – 
ħ²
  
∂²φ
  + Vφ
∂t 2m ∂q²

What if φ 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:

iħ 
∂ρ
  – ρ 
∂S
  = – 
ħ²
 ┌
│
│
│
└
∂²ρ
  + 
2i
  
∂ρ
  
∂S
  + 
iρ
  
∂²S
  – 
ρ
 ┌
│
│
└
∂S
 ┐
│
│
┘		 2  ┐
│
│
│
┘		  + Vρ
∂t ∂t 2m ∂q² ħ ∂q ∂q ħ ∂q² ħ² ∂q

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

 – ρ 
∂S
  = – 
ħ²
 ┌
│
│
│
└
∂²ρ
  – 
ρ
 ┌
│
│
└
∂S
 ┐
│
│
┘		 2  ┐
│
│
│
┘		  + Vρ
∂t 2m ∂q² ħ² ∂q

which, when divided by ρ, can be simplified as

 – 
∂S
  = 
1
 ┌
│
│
└
∂S
 ┐2
│
│
┘		  – 
ħ²
  
∂²ρ
  + V
∂t 2m ∂q 2mρ ∂q²

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

 – 
∂S
  = 
1
 ┌
│
│
└
∂S
 ┐2
│
│
┘		  + V
∂t 2m ∂q

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