# Finding eigenvalues and functions for Hamiltonian (perturbation theory)

I am trying to find the Eigenvalues for the following equation (which comes from the Pauli equation when $$p^2/m^2c^2\ll 1$$): $$i\hbar\frac{d}{dt}\psi=\left[\frac{\vec{p}^2}{2m}-\frac{e}{2mc}\left(\vec{{L}}+2\vec{S}\right)\cdot\vec{{B}}\right]\psi,$$ where $$\vec{S}=\frac{\hbar\vec{\sigma}}{2}$$, $$\vec{{L}}$$ is the angular orbital momentum operator, and $$\vec{B}=B_{0}\hat{z}$$ is a magnetic field in the $$z$$-direction. $$\psi$$ is a two-component Pauli spinor.

I introduce the value of the magnetic field. $$i\hbar\frac{d}{dt}\psi=\left[\frac{\vec{p}^2}{2m}-\frac{e}{2mc}\left({L_{z}}+\hbar\sigma_z\right)B_0\right]\psi.$$

But from here I genuinely do not know how to proceed. I have thought about using that $$L_z=xp_y-yp_x$$, and maybe try to work from there, but I don't really see it.

What would be the best way forward?