Action as the complex phase of the wave function of quantum mechanics

Question

I heard that the action of classical mechanics can be seen as the complex phase of the wave function of quantum mechanics

$$\psi=\rho \exp\left(\frac{iS}{\hbar}\right)\tag1$$

I am more familiar with the stationary state wavefunction of the form

$${\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )e^{-i{Et/\hbar }}}\tag2$$

Equation $(1)$ seems really nice in that it connects classical mechanics (action) to quantum mechancis (wavefunction), but I can not wrap my head around it. I can not find any derivation of it or any physical meaning.

I found only that when we input $(1)$ into the Schrödinger equation and take the classical limit we obtain the Hamilton-Jacobi equation.

What is the physical interpretation of equation $(1)$ and how did we obtain it?

Answer 1

In connecting classical and quantum mechanics, typically the only thing we can ask for is that in the classical limit (which is often formally taken as the limit $\hbar\rightarrow 0$), the predictions of quantum mechanics match the predictions of classical mechanics.

The phase of the wavefunction is not the classical action in general; a general state will be a complicated superposition of many classical paths, for example. However, in the classical limit, if we write $\psi=\rho e^{i S/\hbar}$ and take the $\hbar\rightarrow 0$ limit, we find that the variable $S$ obeys the same equation that the classical action does in the Hamilton-Jacobi equation, as you've seen. This is the connection. This is one manifestation of the WKB approximation.

Historically, this result was very interesting and important, because it connected with Bohr-Sommerfeld quantization, which was a pre-quantum model that allowed one to calculate fine-structure of Hydrogen.

Answer 2

OP's eq. (1) is the WKB Ansatz for the wavefunction $\psi$ in the TDSE, cf. e.g. this Phys.SE post. Here $S$ is Hamilton's principal function; not the action per se.