Application of WKB method to a magnetization problem

[jannyhuggy]

It is not from my howework(due I'm not in the undergrad now), but it seems to be a very easy question I have to know answer to, but I fail to do so.

Homework Statement

I have to go from classical to quantum Hamiltonian via WKB method (and both to solve Schroedinger equation)
It looks like E=-K₁M_z²+K₂M_x²-(H,M). H =(H_x,H_y,H_z) - external magnetic field, constant in time, M²=const. Here M is the magnetization vector.

Homework Equations

I've got the system of equations of motion for classical case
[itex]\dot{M_x}[/itex]=2K₁M_zM_y+H_zM_y-H_yM_z,

[itex]\dot{M_y}[/itex]=-2(K₁+K₂)M_zM_x+H_xM_z-H_zM_x,

[itex]\dot{M_z}[/itex]=2K₂M_xM_y+H_yM_x-H_xM_y

The Attempt at a Solution

Then I need to use WKB method, and I have 2 variables. When I write down the Schroedinger equation I either need to separate variables (and then solve using WKB like hydrogen atom) or to use multiple-variables WKB method if they cannot be separated.

Both in xyz and spherical (if I didn't do any mistake) I cannot separate variables in Schroedinger equation.
Any ideas how to apply WKB here (or ideas of I did mistakes)?

So, the question is
- to help me check whether in any coordinate system variables can be separated, if yes - in which one?
- if they cannot - how to apply 2-variable WKB method here?

 

[mark726]

Thank you for reaching out for help with this problem. It is a common misconception that the WKB method can only be applied to one-dimensional problems. In fact, it can be extended to multi-dimensional problems as well, such as the one you have described here.

In order to apply the WKB method to this problem, you will need to use the WKB approximation for a multi-dimensional system. This approximation involves expanding the wave function in a series of powers of ℏ, and taking into account the derivatives with respect to each variable. This will result in a set of coupled equations, similar to the ones you have already derived for the classical case.

To solve these equations, you will need to use the method of separation of variables. In this case, you will need to find a set of coordinates in which the Hamiltonian can be written as a sum of squares of operators. This will allow you to separate the equations into a set of one-dimensional equations, which can then be solved using the WKB method for one-dimensional systems.

I hope this helps you to successfully solve the problem. Good luck!