How to diagonalize Hamiltonian with Zeeman field

[aaaa202]

Recently I have been asked to solve the problem of an electron in a Zeeman-field that couples the spin of the electron to the magnetic field.
I am not sure how to correctly set up the problem. I think, however, that what I have done on the picture is correct. The usual p^2/2m + V term in the Hamiltonian is tensored with the identity matrix and the Zeeman term is added in the usual way.
I am however unsure how to solve this Hamiltonian. How do you solve something like this with both an eigenvalue equation in the matrix structure and in the spatial part of H?

 

[Demystifier]

This is usually solved perturbatively. Most QM textbooks have a chapter on time-independent perturbation theory.

 

[aaaa202]

Okay but is it not possible to find the exact analytical expression for the eigenenergies?

 

[DrDu]

Easy. Just find a unitary transformation which rotates B in the direction of the z axis. Then the hamiltonian will be diagonal as only the sigma_z component remains.

 

[Demystifier]

Easy. Just find a unitary transformation which rotates B in the direction of the z axis. Then the hamiltonian will be diagonal as only the sigma_z component remains.

This diagonalizes the interaction Hamiltonian, but not the full Hamiltonian.

 

[DrDu]

If B is constant, it does, at least in spin space.