
\sect{Baruch's  B08.}

Consider a two dimensional bose gas in a harmonic potential
with energy eigenvalues $\hbar\omega(n_1+n_2+1)$ where $n_1,\,
n_2$ are integers. [This is how the recent cold atom experiments
realize condensation].
\begin{itemize}
\item[(a)] Show that the average particle number is
\[N=\left(\frac{k_BT}{\hbar \omega}\right)^2g_2(\zeta)+N_0\]
where $N_0$ is the the ground state occupation and $g_2(\zeta)$ as
defined in class. Assume $k_BT\gg \hbar\omega$ so that summations
on $n_1,\, n_2$ can be replaced by integrals.
\item[(b)] Use $g_2(1)=\pi^2/6$ to infer the Bose Einstein condensation
temperature $T_c$. [Note that $N$ here is not taken to its
thermodynamic limit; the transition is still fairly sharp if
$N>>1$.]

\item[(c)] Show that for $T<T_c$: $\,\,N_0=N[1-(T/T_0)^2].$\\
\end{itemize}

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