I'm having some trouble computing ${\rm tr}\left[\hat{a}^{\dagger}\hat{a}\hat{D}(\alpha)\right]$ where $\hat{D}(\alpha)$ is the displacement operator.
I tried doing this using the number state basis, which allows us to write the trace of an operator $\hat{A}$ as $${\rm tr}\left(\hat{A}\right) = \sum_{n \geq 0} \langle n |\hat{A} | n \rangle $$ Using that $\hat{a}^{\dagger}\hat{a}$ sends $|n \rangle \mapsto n |n \rangle$, I got that $${\rm tr}\left[\hat{a}^{\dagger}\hat{a}\hat{D}(\alpha)\right]= \sum_{n \geq 0} n \langle n |\hat{D}(\alpha) | n \rangle,$$ but am unable to continue from here. I was wondering if someone could direct me to another approach or a reference where this is worked out. (For context, I actually want to compute the Wigner function of $\hat{a}^{\dagger}\hat{a}$.)
