# Eigenvalues of a spin-orbit Hamiltonian

Good day everyone,

The question is as following:

Consider an electron gas with Hamiltonian:
$$\mathcal{H} = -\frac{\hbar^2 \nabla^2}{2m} + \alpha (\boldsymbol{\sigma} \cdot \nabla)$$

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:

σ is given by the Pauli matrices:
$$\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$$
$$\sigma_x = \left(\begin{matrix} 0&1\\1&0\end{matrix}\right),$$
$$\sigma_y = \left(\begin{matrix} 0&-i\\i&0\end{matrix}\right),$$
$$\sigma_z = \left(\begin{matrix} 1&0\\0&-1\end{matrix}\right)$$

Attempt at the solution:

I started with calling the eigenvector as following:
$$|\Psi> = Ae^{i\textbf{k}\cdot \textbf{r}}|\psi> ,$$
which need to fulfill the eigenvalue equation:
$$\mathcal{H}|\Psi> = E|\Psi>.$$
This eigenvector gives the following:
$$\mathcal{H}|\Psi> = Ae^{i\textbf{k}\cdot \textbf{r}}\frac{\hbar^2 k^2}{2m}|\psi> +Ae^{i\textbf{k}\cdot \textbf{r}}\alpha i(\boldsymbol{\sigma} \cdot \textbf{k})|\psi> = E\cdot Ae^{i\textbf{k}\cdot \textbf{r}}|\psi>,$$
where
$$\alpha i(\boldsymbol{\sigma} \cdot \textbf{k}) = \left(\begin{matrix} \alpha i k_z & \alpha i k_x + \alpha k_y\\ \alpha i k_x - \alpha k_y & -\alpha i k_z \end{matrix}\right).$$
This can then be reduced to a matrix eigenvalue equation:
\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*}

However, this matrix gives the eigenvalues:
$$E_1 = \frac{\hbar^2k^2}{2m} + \alpha i k \\ E_2 = \frac{\hbar^2k^2}{2m} - \alpha i k$$

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?

Ian Berkman

Paul Colby
Gold Member
The second term in your Hamiltonian doesn't appear to be hermitian? Other than that your steps seem fine.

Do you mean the αi(σ⋅k) term?
I have checked that term couple of times, but I cannot see where it went wrong.

I could try to solve it with the function
$$|\Psi> = A e^{\textbf{k} \cdot \textbf{r}}|\psi>$$

Which gives a Hermitian operator, however, as far as I see, it also gives a negative h2k2/2m term in the energy eigenvalue. Furthermore, I cannot remember having ever seen such a function.

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Nevermind the function in my comment above, it is not right.

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vela
Staff Emeritus
Homework Helper
The spin-orbit term isn't hermitian because of the ##\nabla##. Shouldn't it be written in terms of the momentum, which will supply a factor of ##i##?

blue_leaf77
Paul Colby
Gold Member
Well, you have the right eigenvalues, eigenvalues should be real in this case. Choosing ##\alpha## imaginary would do the trick. I suspect your problem has a typo.

Paul Colby
Gold Member
Do you mean the αi(σ⋅k) term?

Yes, interesting it appears with an ##i## here but not in your original problem statement.

The i appears in the second term because of the gradient of e^(ikr).
I am going to mail the professor if alpha could be an imaginary value, however, I would say that that would be stated in the problem if that is the case.

The spin-orbit term isn't hermitian because of the ##\nabla##. Shouldn't it be written in terms of the momentum, which will supply a factor of ##i##?

The extra factor of ##i## should also appear when ##\alpha## could only take an imaginary value. Also, a momentum operator fits better in a spin-orbit Hamiltonian.

I think indeed this problem has a typo, I will keep you informed about the professor's reply.

The problem was indeed incorrectly stated. Well, sometimes these things happen.

Thank you all for helping.

Ian