Derive Schrodinger equation

[hunc]

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!

 

[stevendaryl]

You can derive it in the following way:

1. Let $\phi = a e^{\frac{iS}{\hbar}}$
2. Compute $\frac{-\hbar^2}{2m} \nabla^2 \phi$ in terms of $a$ and $S$ and only keep the lowest-order terms, in powers of $\hbar$
3. Similarly, compute $i \hbar \frac{d}{dt} \phi$ in terms of $a$ and $S$ and keep the lowest-order terms in powers of $\hbar$.
4. Now, use the fact that $\frac{(\nabla S)^2}{2m} + V = -\frac{d}{dt} S$ to show that the results of 2 and 3 are equal (to lowest order in $\hbar$)

 

[hunc]

You can derive it in the following way:

1. Let $\phi = a e^{\frac{iS}{\hbar}}$
2. Compute $\frac{-\hbar^2}{2m} \nabla^2 \phi$ in terms of $a$ and $S$ and only keep the lowest-order terms, in powers of $\hbar$
3. Similarly, compute $i \hbar \frac{d}{dt} \phi$ in terms of $a$ and $S$ and keep the lowest-order terms in powers of $\hbar$.
4. Now, use the fact that $\frac{(\nabla S)^2}{2m} + V = -\frac{d}{dt} S$ to show that the results of 2 and 3 are equal (to lowest order in $\hbar$)

Thanks! It was my first attempt, which never really got carried out. I thought $\nabla^2\phi$ can bring in $1/\hbar^2$ and $\partial_t \phi$ only $1/\hbar$... And I just go through it and all is fine.
And now I kind of want to ask what's the story about the wiki and the equation
$$
i\hbar\partial_t \phi=-\frac{\hbar^2}{2m}\frac{(\nabla \phi)^2}{\phi}+V(x)\phi.$$

 

[stevendaryl]

Well, if you let $\phi = a e^{\frac{i}{\hbar} S}$, then

$\phi^* (-\frac{\hbar^2}{2m} \nabla^2 \phi) = -\frac{\hbar^2}{2m} a \nabla^2 a + \frac{\hbar^2}{2m} (\nabla a)^2 - \frac{i \hbar}{2m} (\nabla^2 S) a^2 + \frac{1}{2m} (\nabla S)^2 a^2$

$\frac{\hbar^2}{2m} |\nabla \phi|^2 = \frac{\hbar^2}{2m} (\nabla a)^2 + \frac{1}{2m} (\nabla S)^2 a^2$

So the difference between them is $-\frac{\hbar^2}{2m} a \nabla^2 a - \frac{i \hbar}{2m} (\nabla^2 S) a^2$.

 

[paranormal]

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum state over time. It was first derived by Erwin Schrodinger in 1925 and is based on the principles of wave mechanics.

To derive the Schrodinger equation, we start with the Hamilton-Jacobi equation, which is a classical equation that describes the evolution of a classical system. In the Hamilton-Jacobi equation, the quantity S represents the classical action, which is a measure of the energy of the system. The book suggests that we can use the concept of wave mechanics to express S in terms of a wave function, ψ.

Substituting this expression for S into the Hamilton-Jacobi equation, we get an equation that involves the wave function ψ and its derivatives. This equation is known as the Schrodinger equation, and it describes the evolution of the quantum state over time.

The leading term in the Schrodinger equation is the Laplacian operator, which is represented as ∇^2 or Δ. This term represents the kinetic energy of the system and is related to the momentum of the particles in the system. The book suggests that this term can be obtained from the expression for S by taking the leading terms when ℏ is small compared to a typical classical action S.

In summary, the Schrodinger equation is derived by substituting the expression for S in terms of the wave function ψ into the classical Hamilton-Jacobi equation. The leading term in the resulting equation is the Laplacian operator, which represents the kinetic energy of the system. This equation is known as the Schrodinger equation and it describes the evolution of the quantum state over time.