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## Homework Statement

Compute the partition function Z = Tr(Exp(-βH)) and then the average number of particles

in a quantum state <n

_{α }> for an assembly of identical simple harmonic oscillators. The Hamiltonian is:

H = [itex]\sum _{k}[/itex][(n

_{k}+1/2)[itex]\hbar - \mu[/itex] n

_{k}]

with n

_{k}=a

_{k}

^{+}a

_{k}.

Do the calculations once for bosons and once for fermions.

## Homework Equations

## The Attempt at a Solution

For Bosons

[itex]Z=\text{Tr}[\text{Exp}[-\text{$\beta $H}]]=\sum _n \langle n|\text{Exp}[-\text{$\beta $H}]|n\rangle =\sum _n \left\langle n\left|\text{Exp}\left[-\beta

\left(\sum _k \left(n_k+\frac{1}{2}\right)\hbar -\text{$\mu $n}_k\right)\right]\right|n\right\rangle =[/itex]

[itex]=\sum _n \left\langle n\left|\text{Exp}\left[-\beta \left(\sum _k (\hbar -\mu )a_k{}^+a_k-\frac{1}{2}\hbar \right)\right]\right|n\right\rangle[/itex]

I don't know how to go on from here.

Particularly I don't know how to deal with the sum in the exponential.