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\sect{The functions $N(E)$ and $Z(T)$ 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}