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The question is as following:

The question is as following:

Consider an electron gas with Hamiltonian:

[tex] \mathcal{H} = -\frac{\hbar^2 \nabla^2}{2m} + \alpha (\boldsymbol{\sigma} \cdot \nabla)[/tex]

where α parameterizes a model spin-orbit interaction. Compute the eigenvalues and eigenvectors of wave vector k and plot them in the x-direction. Interpret the results.

Relevant equations:

Relevant equations:

σ is given by the Pauli matrices:

[tex] \boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)[/tex]

[tex] \sigma_x = \left(\begin{matrix} 0&1\\1&0\end{matrix}\right),[/tex]

[tex] \sigma_y = \left(\begin{matrix} 0&-i\\i&0\end{matrix}\right),[/tex]

[tex] \sigma_z = \left(\begin{matrix} 1&0\\0&-1\end{matrix}\right)[/tex]

Attempt at the solution:

Attempt at the solution:

I started with calling the eigenvector as following:

[tex]|\Psi> = Ae^{i\textbf{k}\cdot \textbf{r}}|\psi> ,[/tex]

which need to fulfill the eigenvalue equation:

[tex]\mathcal{H}|\Psi> = E|\Psi>. [/tex]

This eigenvector gives the following:

[tex] \mathcal{H}|\Psi> = Ae^{i\textbf{k}\cdot \textbf{r}}\frac{\hbar^2 k^2}{2m}|\psi> +Ae^{i\textbf{k}\cdot \textbf{r}}\alpha i(\boldsymbol{\sigma} \cdot \textbf{k})|\psi> = E\cdot Ae^{i\textbf{k}\cdot \textbf{r}}|\psi>,[/tex]

where

[tex] \alpha i(\boldsymbol{\sigma} \cdot \textbf{k}) = \left(\begin{matrix} \alpha i k_z & \alpha i k_x + \alpha k_y\\ \alpha i k_x - \alpha k_y & -\alpha i k_z \end{matrix}\right).[/tex]

This can then be reduced to a matrix eigenvalue equation:

[tex]\begin{align*}E|\psi> &= \left(\frac{\hbar^2 k^2}{2m} +\alpha i (\boldsymbol{\sigma} \cdot \textbf{k})\right)|\psi>\\ E|\psi> &= \left(\begin{matrix} \frac{\hbar^2 k^2}{2m} + \alpha i k_z & \alpha i k_x + \alpha k_y \\ \alpha i k_x - \alpha k_y & \frac{\hbar^2 k^2}{2m} - \alpha i k_z \end{matrix}\right)|\psi>\end{align*}[/tex]

However, this matrix gives the eigenvalues:

[tex] E_1 = \frac{\hbar^2k^2}{2m} + \alpha i k \\

E_2 = \frac{\hbar^2k^2}{2m} - \alpha i k [/tex]

Which look quite right, except that they contain imaginary parts which suggest some form of energy loss/damping which is not stated in the problem.

Did anybody see where I went wrong?

Thanks in advance,

Ian Berkman