Finding the spectrum of a curious hamiltonian

Question

I wish to analyse the following hamiltonian, i.e. find its eigenvalues and eigenstates.

$$H = \frac{1}{2}\epsilon(\sigma _z \otimes \mathbb{1} + 1\otimes \sigma _z) - \Delta (\sigma _x \otimes \sigma _x)$$

Where the ones denoted identity operators on $H_2$ (2-dim Hilbert space), and the sigmas are the Pauli matrices. The epsilon and delta are positive constants.

First of all, I'm curious what sort of interaction is represented by this hamiltonian. It appears to act on $H_2 \otimes H_2$, so some sort of system composed of two particles, with two states each, and the appearance of the Pauli matrices screams "spin". So I'm guessing it's some sort of spin interaction. Not sure what the mysterious constants $\epsilon, \Delta$ would represent, though the letters chosen would indicate that one is probably gonna be thought of as small, and the other as large. Maybe some sort of perturbation expansion? I'd like to learn how to analyse the spectrum of such an operator.

Answer 1

Your Hilbert space is finite-dimensional (specifically, 4-dimensional). It means that you need no fancy way of deriving the spectrum, just proceed with the standard approach:

1. Write your Hamiltonian in the matrix form and solve the characteristic equation $ \det ( H - \lambda \cdot 1_{4 \times 4} ) = 0 $ with respect to $\lambda$, which would give you eigenvalues of $H$ (the energy spectrum).

2. For each eigenvalue $\lambda$, solve the linear system $H \psi = \lambda \psi$ which would give you the eigenspace of eigenstates corresponding to $\lambda$.

I tried to post a detailed answer here, but apparently it was deleted because of the homework policy.

P.S. no perturbation expansion is needed.

Answer 2

Your Hilbert space is finite-dimensional (specifically, 4-dimensional). It means that you need no fancy way of deriving the spectrum, just proceed with the standard procedure. First, write your Hamiltonian in the matrix form:

$$ H = \left( \begin{array}{cccc} \epsilon & -\Delta & 0 & 0 \\ -\Delta & 0 & 0 & 0 \\ 0 & 0 & \epsilon & -\Delta \\ 0 & 0 & -\Delta & 0 \\ \end{array} \right). $$

You can immediately see that your system is actually a direct sum of two independent systems with hamiltonians

$$ h = \left( \begin{array}{cc} \epsilon & -\Delta \\ -\Delta & 0 \\ \end{array} \right). $$

But let me proceed as if I don't know that.

You can solve the characteristic equation $\det \left( H - \lambda \cdot 1_{4 \times 4} \right) = 0$ with respect to $\lambda$ which would give you the eigenvalues (energy spectrum) of $H$:

$$ \left\{\frac{1}{2} \left(\epsilon -\sqrt{4 \Delta ^2+\epsilon ^2}\right),\frac{1}{2} \left(\sqrt{4 \Delta ^2+\epsilon ^2}+\epsilon \right)\right\}. $$

Both eigenstates are degenerate of order 2 (because they are roots of order 2 of the characteristic equation).

The corresponding eigenstates are:

$$ \left\{\left( \begin{array}{c} 0 \\ 0 \\ \frac{\sqrt{4 \Delta ^2+\epsilon ^2}-\epsilon }{2 \Delta } \\ 1 \\ \end{array} \right),\left( \begin{array}{c} \frac{\sqrt{4 \Delta ^2+\epsilon ^2}-\epsilon }{2 \Delta } \\ 1 \\ 0 \\ 0 \\ \end{array} \right)\right\} $$

(for the first eigenvalue) and

$$ \left\{\left( \begin{array}{c} 0 \\ 0 \\ -\frac{\epsilon +\sqrt{4 \Delta ^2+\epsilon ^2}}{2 \Delta } \\ 1 \\ \end{array} \right),\left( \begin{array}{c} -\frac{\epsilon +\sqrt{4 \Delta ^2+\epsilon ^2}}{2 \Delta } \\ 1 \\ 0 \\ 0 \\ \end{array} \right)\right\} $$

for the second one.

No perturbation expansion is needed! You can build an arbitrary state time evolution out of the energy eigenstates via the general formula:

$$ \Psi = \sum e^{-i E_a t / \hbar} \psi_a $$

Note that $\epsilon$ and $\Delta$ have to be real in order for $H$ to be hermitian.