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\sect{Spectral functions for N spins}
Consider an ${N}$ spin system:
\[\hat{H}=\sum_{\alpha=1}^{N}\,\frac{\varepsilon}{2}\,\hat{\sigma}_{z}^{\left(\alpha\right)}\]
Calculate ${Z_{N}\left(\beta\right)}$ in two different ways:
\begin{itemize}
\item[(1)]
The short way - Calculate ${Z_{N}\left(\beta\right)}$ by factoring the sum.
\item[(2)]
The long way - Write the energy levels ${E_{n}}$ of the system.
Mark with ${n=0}$ the ground level, and with ${n=1,2,3,...}$ the
excited levels. Find the degeneracy ${g_{n}}$ of each level.
Use these results to express $Z_{N}\left(\beta\right)$,
and show the that the same result is obtained.
\end{itemize}