# Density matrix in the canonical ensemble

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1. Nov 14, 2014

### mdk31

1. The problem statement, all variables and given/known data

We have a quantum rotor in two dimensions with a Hamiltonian given by $$\hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2}$$. Write an expression for the density matrix $$\rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle$$

2. Relevant equations
$$\hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2}$$
$$\rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle$$

3. The attempt at a solution

In the canonical ensemble, I know that $$\hat{\rho}$$ is given by:
$$\hat{\rho} =\dfrac{1}{Z} e^{-\beta \hat{H}}$$ where Z is the partition function. But this is about as far as I can get. Any assistance towards a solution would be greatly appreciated.

2. Nov 20, 2014