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## Homework Statement

I am reading

*Mathematical Concepts of Quantum Mechanics (Stephen J. Gustafson, Israel Michael Sigal. Second edition)*. The book would like to find an evolution equation which would lead to the Hamilton-Jacobi equation

$$\frac{\partial S}{\partial t}=-h(x, \nabla S) $$

in the way the wave equation led to the eikonal one. The book also says that ##\phi (x, t) = a(x, t) \exp( i S(x,t)/\hbar)##. So I express ##S(x,t)## using ##\phi (x,t)## and substitute back to the Hamilton-Jacobi equation, taking ##h (x, \nabla S) = \frac{1}{2m}|\nabla S|^2+V(x)##.

The book means to take the leading terms when ##\hbar## small compared to a typical classical action ##S## and restore Schrodinger equation. I am kind of lost during the derivation.

## Homework Equations

After the substitution, I have

$$i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}[(\frac{\nabla \phi}{\phi}-\frac{\nabla a}{a})^2-\frac{2im\partial_t a}{a\hbar}]\phi+V(x)\phi.$$

Comparing with Schrodinger equation, I figure that the leading term of

$$[(\frac{\nabla \phi}{\phi}-\frac{\nabla a}{a})^2-\frac{2im\partial_t a}{a\hbar}]\phi$$

should equal to ##\Delta_x \phi##, but don't know how.

## The Attempt at a Solution

I am not sure what to search for the problem, but wiki have something on this. A

*nonlinear*variant of the Schrödinger equation is expressed as

$$i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}\frac{(\nabla \phi)^2}{\phi}+V(x)\phi.$$

I am not sure what a

*nonlinear Schrödinger equation*is after realizing it's not the same thing as the Schrödinger equation.

The book's goal seems to be the linear Schrödinger equation. Even though I do see how to obtain the

*nonlinear Schrödinger equation,*I am not sure why ##(\frac{\nabla \phi}{\phi})^2## is a leading term. Could someone help me with this?

Thanks!