# Homework Help: Single Mode Thermal Field

1. Dec 5, 2017

### BOAS

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

I am having trouble connecting the expectation value of $\hat a^\dagger \hat a$ to the prediction from statistical mechanics for the state $$\hat \rho = \sum_n \frac{\bar n^n}{(1 + \bar n)^{n+1}} |n\rangle \langle n|$$

2. Relevant equations

3. The attempt at a solution

from statistical mechanics, we have that $\bar n = \frac{1}{e^{\beta \hbar \omega} -1}$ and so I try to compute this by taking $\langle \hat a^\dagger \hat a \rangle$.

$$\langle\hat n\rangle = Tr(\rho \hat n)$$

$$\langle \hat n \rangle = Tr(\sum_n \frac{\bar n^n}{(1 + \bar n)^{n+1}} |n\rangle \langle n| \hat n)$$

$$\langle \hat n \rangle = \sum_n \frac{\bar n^n}{(1 + \bar n)^{n+1}}\langle n| \hat n |n\rangle$$

$$\langle \hat n \rangle = \sum_n \frac{n \bar n^n}{(1 + \bar n)^{n+1}}$$

I am unsure of where to go from here, or if I am approaching this in the correct manner.

Am I supposed to be able to evaluate this sum?

Edit - So I typed this into mathematica and found that the sum does indeed converge to $\bar n$. I suppose my question is really, how should I go about evaluating this sum?

Last edited: Dec 5, 2017
2. Dec 10, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Dec 11, 2017

### BOAS

This thread can be closed, I have solved it.