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\begin{document}
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\heading{E0105: The functions $N(E)$ and $Z(T)$ for a particle in a double well}
\auname{Shira Wurzberg}
{\bf The problem:}
\Dn
\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}
\Dn\Dn
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{\bf The solution:}
\Dn
(a) The trajectories of a particle in a confining potential are those of constant energy
\begin{align}
E = \frac{p^2}{2m} + V(x) \qquad \rightarrow \qquad p(x) = \pm \sqrt{2m\left(E - V(x)\right)}.
\end{align}
Therefore for $|E| \leq \frac{\epsilon}{2}$ we have
\begin{align}
0 < x < \frac{L}{2} \qquad \qquad p(x) = \pm \sqrt{2m\left(E + \frac{\epsilon}{2}\right)},
\end{align}
while for $E > \frac{\epsilon}{2}$ we have
\begin{align}
0 < x < L \qquad \qquad p(x) = \left\{ \begin{array}{lc}
\pm \sqrt{2m\left(E + \frac{\epsilon}{2}\right)} & 0 < x \leq \frac{L}{2} \\
\pm \sqrt{2m\left(E - \frac{\epsilon}{2}\right)} & \frac{L}{2} < x < L
\end{array} \right. \nonumber
\end{align}
% === Figure >>>
\begin{figure}[htb]
\begin{center}
\vspace{0cm}
\includegraphics[width=10cm,angle=0]{0105_trajectories.png}
\vspace{0cm} \caption{ \label{Fig:CondCorrections} %
The possible trajectories in the phase space for a particle in the given well for (a) $|E| \leq \frac{\epsilon}{2}$ and (b) $E > \frac{\epsilon}{2}$. }
\end{center}
\end{figure}
\Dn
(b) In the semi-classical approximation $x$ and $p$ are taken to be independent variables, and a ``state'' in phase space is of volume $(2\pi\hbar)^d$.
$N(E)$ are the number of states with energy smaller than $E$,
\begin{align}
N(E) = \int \frac{d^d x d^d p}{(2\pi\hbar)^d} \Theta(E - H(x,p)),
\end{align}
where $\Theta(x)$ is the step function. Thus according to the above,
\begin{align}
N\left(|E| < \frac{\epsilon}{2}\right) &= \frac{L/2}{2\pi\hbar} 2 \sqrt{2m\left(E + \frac{\epsilon}{2}\right)} = \frac{L}{2\pi\hbar} \sqrt{2m\left(E + \frac{\epsilon}{2}\right)} \\
N\left(E > \frac{\epsilon}{2}\right) &= \frac{L}{2\pi\hbar} \left[ \sqrt{2m\left(E + \frac{\epsilon}{2}\right)} + \sqrt{2m\left(E - \frac{\epsilon}{2}\right)} \right].
\end{align}
\Dn
(c) The partition function $Z(\beta)$ is given by
\begin{align}
Z(\beta) &= \int_{-\infty}^{\infty} \frac{dx dp}{2\pi\hbar} e^{-\beta ( p^2/2m + V(x) )} = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dp e^{-\beta p^2/2m} \times \int_{-\infty}^{\infty} dx e^{-\beta V(x)} \nonumber \\
&= \frac{1}{2\pi\hbar} \sqrt{\frac{2m\pi}{\beta}} {L \over 2} \left( e^{\beta {\epsilon \over 2}} + e^{-\beta {\epsilon \over 2}} \right) = {1 \over \hbar} \sqrt{\frac{m}{2\pi\beta}} L \cosh\left( \beta {\epsilon \over 2} \right).
\end{align}
We can see that $Z$ can be expressed as (kinetic term)$\times$(spin term).
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\end{document}