# Discrete 3-particle system - Condensed Matter Pysics

1. The problem statement, all variables and givenknown data
We have a system of three atoms arranged in a circular arrangements. They each have a valence electron that can tunnel to the nearest neighbor. For a tunneling rate $-A/\hbar$ we have the Hamiltonian (shifted by an energy $E_a$ on the diagonal)
$$H = \begin{pmatrix} 0 & -A & -A\\ -A & 0 & -A\\ -A & -A & 0 \end{pmatrix}$$
which eigenvalues $-2|A|$ corresponding to the eigenstate (1,1,1) and degenerated eigenvalues $A$ corresponding to eigenstates (-1,0,1) and (-1,1,0). Since we have translational invariance, $p$ is a good quantum number, hence $[H,p]=0$ and $p$ and $H$ can be diagonalized simultaneously. Assuming plane wave solutions $\psi_n \sim \mathrm{e}^{ikn}$ we get from boundary conditions that $k = 2/3 \pi N$ for $N \in \mathbb{Z}$. Now, the three lowest cases are $k=0$ and $k= \pm 2/3 \pi$. $k=0$ corresponds to the eigenvalue $-2|A|$ since for $k=0$ we have $\psi_1=\psi_2=\psi_3=1$, i.e. the electrons are evenly distributed over the three atoms. The other two cases give the following linear combinations:
$k=2/3 \pi$: $$|\psi_2 \rangle + (-1/2 -i \sqrt{3}/2)|\psi_3 \rangle$$
$k=-2/3 \pi$: $$|\psi_2 \rangle + (-1/2 +i \sqrt{3}/2)|\psi_3 \rangle$$

My question is now: How do I find the momentum values $p$ for each of the 3 simultaneous eigenstates?

## Homework Equations

$\hat{p} | \psi_n \rangle = p | \psi_n \rangle$

## The Attempt at a Solution

I'm really at a loss here. How can we find the momentum values for this discrete system with only assumed plane wave solutions?

Since we have translational invariance, $p$ is a good quantum number, hence $[H,p]=0$ and $p$ and $H$ can be diagonalized simultaneously. Assuming plane wave solutions $\psi_n \sim \mathrm{e}^{ikn}$ we get from boundary conditions that $k = 2/3 \pi N$ for $N \in \mathbb{Z}$.