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I'm trying to understand more about the correspondence between classical and quantum physics, in particular the relation between classical mechanics in Hamilton-Jacobi form and the Schrödinger equation. The former can be obtained from quantum theory by plugging

[tex]\psi=\psi_0 e^{\frac{i}{\hbar}S}[/tex]

into the Schrödinger equation, which yields

[tex]\frac{1}{2m}(\nabla S)^2 + \frac{\partial S}{\partial t} - \frac{i \hbar}{2m}\Delta S=0,[/tex]

where the last term vanishes in the classical limit. My first question is: What is the interpretation of this quantum term? How is the classical theory affected by it?

Then, one can also start from the classical Hamilton-Jacobi equation and, with the help of the wave function [tex]\psi[/tex] from above, rewrite it as

[tex]\frac{\hbar^2}{2m\psi}(\nabla \psi)^2 + i\hbar\frac{\partial \psi}{\partial t} = 0. [/tex]

In this light, one obtains the quantum equation by "replacing the square of the derivative with the second derivative", citing a phrase I sometimes came across. So my second question is: How can this nonlinear variant of the Schrödinger equation be interpreted, how is its behaviour different from the full quantum theory?

In summary: What is the interpretation of the operation "replace square of derivative with second derivative" ?