- #1
IanBerkman
- 52
- 1
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 \\<br /> 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?
Thanks in advance,
Ian Berkman
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 \\<br /> 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?
Thanks in advance,
Ian Berkman