What is the Wigner function of a thermal state? I am wondering how you would compute the Wigner Function of a Thermal State with
average phonon number $\bar{n}_{\mathrm{th}}$.
I know the result should be a Gaussian with variance in position $\langle x^2\rangle = (2 \bar{n}_\mathrm{th}+1) x_\mathrm{zp}^2$ and in momentum $ \langle p^2\rangle = (2 \bar{n}_\mathrm{th}+1) \hbar/(4x_\mathrm{zp}^2)$.
But how do I show that?
I write the thermal density matrix in the Fock basis:
\begin{equation}
\rho_\mathrm{th} = \sum_n \frac{\bar{n}_\mathrm{th}^n}{(1+\bar{n}_\mathrm{th})^{n+1}}|n\rangle \langle n | 
\end{equation}
and use the Wigner Tranform:
\begin{equation}
W_\mathrm{th}(x,p) = \int du \langle x- u/2 | \rho_\mathrm{th} | x + u/2 \rangle e^{\mathrm{i} p u/\hbar} 
\end{equation}
After inserting the density matrix into the Wigner transform I get:
\begin{equation}
W_\mathrm{th}(x,p) = \sum_n \frac{\bar{n}_\mathrm{th}^n}{(1+\bar{n}_\mathrm{th})^{n+1}} \int du \langle x- u/2  |n\rangle \langle n  | x + u/2 \rangle e^{\mathrm{i} p u/\hbar} = \sum_n \frac{\bar{n}_\mathrm{th}^n}{(1+\bar{n}_\mathrm{th})^{n+1}} W_n(x,p),
\end{equation}
where $W_n(x,p)$ is the Wigner functions of the n-Fock state given by:
\begin{equation}
W_n(x,p) = \frac{2}{\hbar \pi}(-1)^n e^{-2 \frac{H}{\hbar \omega} } L_n(4 \frac{H}{\hbar \omega} ),
\end{equation}
with $H= \frac{1}{2} m \omega^2 x^2 + \frac{p^2}{2 m}$, and $L_n$ the nth Laguerre poloynomial.
Everything correct until here?
Now I am too stupid to do the last sum. Was looking for identities and what not half of this day.
Any ideas? I would also appreciate a simpler solution. Thanks for any Help!
 A: I believe the correct Wigner function for the eigenstates is half yours, so take it to be 
\begin{equation}
W_n(x,p) = \frac{(-1)^n}{\hbar \pi} e^{- z/2} L_n(z ),
\end{equation}
where $z=4 H/\hbar\omega  $. 
You know that, since the resolution of the identity must be
$$
\sum_n  W_n= \frac{1}{2\pi \hbar}=1/h,
$$
which, indeed, holds (trivially checkable)  by dint of the standard generating function of the Laguerre polynomials, 
$$
\sum_n t^n L_n(z)= \frac{e^{-tz/(1-t)}} {1-t}  ~~.
$$
Your sum then readily collapses to 
$$
 \sum_n \frac{\bar{n}_\mathrm{th}^n}{(1+\bar{n}_\mathrm{th})^n} W_n(x,p)= 
\frac{e^{-z/2}}{\pi \hbar}  \sum_n \left (\frac{- \bar{n}_\mathrm{th}}{1+ \bar{n}_\mathrm{th}} \right )^n  L_n  (z) = \frac{ (1+\bar{n}_\mathrm{th})}{\pi \hbar  (1+2\bar{n}_\mathrm{th})} ~ e^{-z / 2 (1+2 \bar{n}_\mathrm{th}) } ,
$$
a gaussian in x and p with the requisite widths.
These are the basic maneuvers in phase-space quantization,  Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014.
