Partition function of harmonic oscillator with additional force

[brother toe]

Homework Statement

Show that the partition function for the harmonic oscillator with an additional force $H = \hbar \omega a^{\dagger} a - F x_0 (a + a^{\dagger})$ is given by $\frac{e^{\beta \frac{F^2 x_{0}^2}{\hbar \omega}}}{1-e^{\beta \hbar \omega}}$ and calculate $\left<x\right> = x_0 \left<a + a^{\dagger}\right>$.

Homework Equations

The Attempt at a Solution

The partition function is given by $\sum_{i} e^{-\beta E_i}$ but I am struggling to find the eigenvalues of the Hamiltonian. In pertubation theory the additional terms would not contribute, so the partition function would be the same as the normal harmonic oscillator, but since F is not given as particularly small we cannot use pertubation. I rewrote the Hamiltonian in terms of x and p operators, but I could not solve the resulting differential equation.

I would very much appreciate any help

 

[vela]

Your Hamiltonian is
$$H = \frac{p^2}{2m} + \frac 12 m\omega^2 x^2 - Fx.$$ Complete the square and then change variables to get rid of the offset in $x$.