Let $a^{\dagger}_{\alpha}$ , $a_{\alpha}$ be Fermion creation and annihilation operators, and assume

\begin{displaymath}H_{o} = \sum_{\alpha} \epsilon_{\alpha} a^{\dagger}_{\alpha}a_{\alpha}.\end{displaymath}

(i) Compute $Tr[e^{\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha}]$

(ii) Compute Tr [ $e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}$].

(iii) Compute $Tr [e^{-\beta H_{o}} a_{\alpha}a^{\dagger}_{\alpha}]$. Using the identity $e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} = -T \frac{\partial}{\partial \epsilon_{\alpha}} e^{-\beta H_{o}}$ check your answer to (i).

(iv) Confirm that the expectation

\begin{displaymath}
\langle n_{a} \rangle = Z^{-1} Tr[e^{-\beta H_{o}} a^{\dagger}_{\alpha} a_{\alpha} ] = f(\epsilon_{\alpha})
\end{displaymath}

where $f(x)$ is the Fermi function.

(v) Repeat the above procedure assuming that $a^{\dagger}_{a}$ and $a_{a}$ are boson operators. What are the restrictions on the values of $\epsilon_{a}$ and what happens when one of these energies $\epsilon_{a}$ becomes zero?

(vi) Calculate


\begin{displaymath}
\left(
\begin{array}{l}
a^{\dagger}_{\alpha}(\tau) \\
a^{}_...
...r}_{\alpha} \\
a^{}_{\alpha}
\end{array}\right)
e^{-\tau H_o}
\end{displaymath}