Single Mode Thermal Field

[BOAS]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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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?

Thank you for any help you can give

 

[BOAS]

This thread can be closed, I have solved it.