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\begin{document}
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\heading{E150: The functions $N(E)$ and $Z(T)$ for ${N}$ spins}
\auname{Elkana Porat}
{\bf The problem:}
\Dn
Given an ${N}$ spin system
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\[\hat{H}=\sum_{\alpha=1}^{N}\,\frac{\varepsilon}{2}\,\hat{\sigma}_{z}^{\left(\alpha\right)}\]
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Calculate the partition function ${Z_{N}\left(\beta\right)}$ 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 basic level and with ${n=1,2,3,...}$ the
following levels. Find the degeneracy ${g_{n}}$ of each level.
Use these results to calculate $Z_{N}\left(\beta\right)$.
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%\[Z_{N}\left(\beta\right)=\sum_{n} g_{n}e^{-\beta E_{n}}\]
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\end {itemize}
% Give a physical meaning to the index ${n}$.
%note: in this question ${n}$ is the index of the energy levels and
%not of the state energies. therefore, there's a need to explicitly
%conclude ${g_{n}}$ in the sum.
\Dn\Dn
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{\bf The solution:}
\Dn
(1) The partition function for a single spin ($E_\downarrow = 0,\ E_\uparrow = \varepsilon$):
\[ Z_1\left(\beta\right) = 1+e^{-\beta\varepsilon} \]
For N spins the partition function is given by:
\[Z_{N}\left(\beta\right)=\sum_{\left\{E^{\left(\alpha\right)}\right\}} e^{-\beta \sum_{\alpha} E^{\left(\alpha\right)}}\]
When ${\alpha}$ indicates one spin and ${\left\{E^{\left(\alpha\right)}\right\}}$ corresponds to a particular configuration of the system.
That may seem complicated, but since the spins do not interact we can factorize the sum and get:
\[ Z_N\left(\beta\right) = \left(1+e^{-\beta\varepsilon}\right)^N \]
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\Dn
(2) Each spin contributes to the total energy level 0 for spin down and ${\varepsilon}$ for spin up. The total energy for a state with ${n}$ spins up will than be: ${E_n = n\varepsilon}$.
The degeneracy for such energy level is the number of different groups of ${n}$ spins up, out of the total $N$ spins in the system. That is also noted as
\[{g\left(n\right) = \left(\begin{array}{c} N \\ n \end{array} \right)}\]
the binomial coefficient.
Now, the partition function is a sum over all the states of the system, which can be written as a sum over all the \textbf{energy levels}, including their degeneracy. In other words:
\[ {Z_N\left(\beta\right) = \sum_{n=0}^{N}{g\left(n\right)e^{-\beta E_n}} = \sum_{n=0}^{N}{\left(\begin{array}{c} N \\ n \end{array} \right)
e^{-\beta n\varepsilon}} }\]
Which we recognize as Newton's binomial expansion for:
$${Z_N\left(\beta\right) = \left(1+e^{-\beta\varepsilon}\right)^N}$$
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\end{document}