# Quadrature distribution for an optical mode in the coherent state

proton4ik
Hey there, the task I'm working on is written below.

Find the quadrature distribution ρ(q), for an optical mode being in the coherent state |α>.
Hint: use ∑Hn
(x)*(t^n)/(n!)

I really am struggling with this type of tasks :D
I tried to follow a solved example that I found in my workbook, but there are no explanations and I'm really not sure if I do anything correctly.

$$\rho(q)=<q|\hat{\rho}|q>=\sum_{n=0}^{\infty} <q| \hat{\rho} {{\psi}_n}^* |n> = \frac {1} {1+n_{th}} \sum_{n=0}^{\infty} (\frac{n_{th}}{1+n_{th}})^n {{\psi}_n}^* |{\psi}_n (q)|^2=\frac{e^{\frac{-q^2}{1+2n_{th}}}}{{\pi (1+2n_{th})}^{1/2}}$$

I have five more exercises like that, but I can't understand the concept. Any help is much appreciated!