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

Consider a particle that has a mass $m$ in a double well. 
The potenial $V(x)$ of the well is described in the figure.

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\begin{itemize}

\item[(a)]
Describe the possible trajectories of the particle in the double well.

\item[(b)]
Calculate ${N(E)}$ and the energy levels in the semi-classical approximation.

\item[(c)]
Calculate ${Z(\beta)}$ and show that it can be written 
as a product of "kinetic" term and "spin" term.

\end{itemize}