(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

I am given the hint to use the ansatz

$$\psi = e^{ik_x x} e^{ik_y y} (\phi_1 \hat{x} + \phi_2 \hat{y})$$

3. 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 [itex]\theta[/itex] the angle [itex]\vec{k}[/itex] makes with the x-axis.

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# Homework Help: Eigenstates of Rashba Spin-Orbit Hamiltonian

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