Eigenstates of Rashba Spin-Orbit Hamiltonian

[korialstasz]

Homework Statement

I am given the Rashba Hamiltonian which describes a 2D electron gas interacting with a perpendicular electric field, of the form
$$H = \frac{p^2}{2m^2} + \frac{\alpha}{\hbar}\left(p_x \sigma_y - p_y \sigma_x\right)$$
I am asked to find the energy eigenvalues and corresponding spinor wavefunctions

Homework Equations

I am given the hint to use the ansatz
$$\psi = e^{ik_x x} e^{ik_y y} (\phi_1 \hat{x} + \phi_2 \hat{y})$$

The Attempt at a Solution

I have diagonalized the Hamiltonian and found the energies to be
$$E = \frac{\hbar^2k^2}{2m} \pm \alpha k$$
But I am at a loss how to proceed with finding the eigenspinors. I don't even really understand the hint, since the hatted vectors aren't spinors at all. I have seen the solutions in papers but cannot find how to actually solve for them. The solutions have the form
$$e^{i\mathbf{k}\cdot\mathbf{x}} \left(
\genfrac{}{}{0pt}{}{1}{\pm i e^{i\theta}}
\right)$$
where \theta the angle \vec{k} makes with the x-axis.

 

[eys_physics]

My understanding is that spinor in this case means eigenvector. So, you should find the eigenvectors corresponding to the energy eigenvalues which you already computed. In other words finding the unknown numbers $\phi_1$ and $\phi_2$.