
\sect{Bose in 2D harmonic trap}

% Baruch's  B08.
% revised by dcohen

Consider a two dimensional bose gas in a harmonic potential
with energy eigenvalues $(1+n_1+n_2)\omega$, where ${n_1, n_2}$ are integers. 
This reflects a conventional setup in actual experiments.
Assume that the temperature $T$ is below the Bose-Einstein condensation temperature $T_c$. 

\begin{itemize}

\item[(a)] Find the average number $N_e(T)$ of particles in the excited states.
Assume ${T \gg \omega}$ so that summations can be replaced by integrals.

\item[(b)] Given that the total number of particles is $N$ 
what is the Bose-Einstein condensation temperature $T_c$. 

\item[(c)] Deduce that the number of condensed particles is $n_0 = N[1-(T/T_c)^2]$

\end{itemize}


\[ \int_{0}^{\infty}\frac{x}{ e^{x} -1} dx =\frac{\pi^2}{6} \] 

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