I am looking for a closed form of the density operator of the quantum harmonic oscillator in thermal equilibrium, preferably in position representation. I am fairly sure it looks like a coherent state but I couldn't find it in any of my books or online sources that I skimmed. If my memory serves me well, then there exists a closed form Wigner function for it too, so I would expect that there also exists a simple pure position representation. But please inform me if I am wrong, if there is no closed form.
A short derivation is also welcome.
EDIT
I have written down what i managed to do. The density operator for a canonical ensemble with $\beta = \frac{1}{k_BT}$ should be $ \hat \rho = \exp(-\beta \hat H)$.Using the resolution of identity, we can write it as $$\begin{aligned} \hat \rho &=\exp(-\beta \hat H)\sum_{n=0}^\infty |n\rangle \langle n|\\ &= \sum_{n=0}^\infty \exp(-\beta E_n)|n\rangle \langle n|\\ &=\sum_{n=0}^\infty \exp\left(-\beta \left(\frac{1}{2}\hbar\omega(n+1) \right) \right)|n\rangle \langle n|\\ \end{aligned}$$ The eigenfunctions of the harmonic oscillator are $$ \langle x | n \rangle =\phi_n(x) = \frac{1}{\sqrt{2^nn!}} \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}}\exp(-\frac{m\omega x^2}{2\hbar})H_n\left(\frac{m\omega}{\hbar} x \right) $$
The position representation of the density operator is then $$\begin{aligned} \langle x|\hat \rho|x'\rangle &= \sum_{n=0}^\infty \exp\left(-\beta \left(\frac{1}{2}\hbar\omega(n+1) \right) \right)\langle x|n\rangle \langle n|x'\rangle\\ &= \sum_{n=0}^\infty \exp\left(-\beta \left(\frac{1}{2}\hbar\omega(n+1) \right) \right)\phi_n(x) \phi^*_n(x')\\ \end{aligned}$$ Can this sum be simplified ?