How to solve Rashba and Dresselhaus SOC Hamiltonian

[drFredkin]

Homework Statement

How can I solve this hamiltonian equation?
H= p²/2m +α/ħ (σ_xp_y - σ_yp_x) + γ/ħ (σ_xp_x - σ_yp_y) + β_xσ_x + β_yσ_y

Homework Equations

Rashba coupling effect equation: H_R=α(σ_yk_x - σ_xk_y)

The Attempt at a Solution

H₀=ħ²k²/2m + H_R+ H_D

 

[hilbert2]

I first thought about solving the eigenvalues and position representation eigenfunctions by assuming the eigenfunctions are of product form $\psi(x,y) = \alpha(x)\beta(y)$ and using separation of variables to get ODE:s for $\alpha$ and $\beta$, but if the coefficients $\sigma_{i}$ are matrices instead of numbers, that probably makes the situation more complicated...