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.

1. The problem statement, all variables and given/known data

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.


2. Relevant 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

3. 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?