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\sect{The functions $N(E)$ and $Z(T)$ for a particle in a double well}

Given particle in a well ${H=\frac{p^{2}}{2m}+ V\left(x\right)}$.
\begin {itemize}

\item[(a)]
Draw in the phase space the possible trajectories of the particle in
the well.

\item[(b)]
Calculate ${\mathcal{N}\left(E\right)}$ and the energy levels in the
semy-classic proximity.

\item[(c)]
Calculate ${Z\left(\beta\right)}$ and show that
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\[Z\left(\beta\right)=\left(\frac{m}{2\pi\beta}\right)^{\frac{1}{2}}L
\cosh \left(\frac{1}{2}\beta E\right)\]


\end {itemize}




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