
\sect{BEC in harmonic potential}

%Baruch's  B11.

The current experimental realizations of Bose Einstein
condensation rely on trapping
cold atoms in a potential. Close to its minimum, the potential can
be expanded to second order, and has the form
\[U({\vec r})=\half m\sum_{\alpha}\omega_{\alpha}^2x_{\alpha}^2\]
where $\alpha=1,...,d$, d is the space dimensionality and the
trapping potential may have different frequencies
$\omega_{\alpha}$ in different directions.
\begin{itemize}
\item[(a)] We are interested in the limit of wide traps such that
$\hbar\omega_{\alpha}\ll k_BT$, and the discreteness of the
allowed energies can be largely ignored. Show that in this limit,
the number of states $N(E)$ with energy less than or equal to $E$,
and the density of states $\rho(E)=dN(E)/dE$ are given by
\[N(E)=\frac{1}{d!}\prod_{\alpha=1}^{d}\left(\frac{E}
{\hbar\omega_{\alpha}}\right) \qquad \Rightarrow \qquad
\rho(E)=\frac{1}{(d-1)!}\frac{E^{d-1}}{\prod_{\alpha=1}^{d}
\hbar\omega_{\alpha}}\] [Hint: The volume of the hyper-pyramid
defined by $\sum_{i=1}^d x_i \leq R$ and $x_i \geq 0$, in d
dimensions is $R^d/d!$ .]

\item[(b)] Show that in a grand canonical ensemble, the number of
particles in the trap is
\[\langle N\rangle =g_d(\zeta)\prod_{\alpha=1}^{d}\left(\frac{k_BT}
{\hbar\omega_{\alpha}}\right)\] where $g_n(\zeta)$ is the usual
Bose function.
\item[(c)] Find the chemical potential in the
high temperature limit.
\item[(d)] Find the temperature $T_c$ for BE condensation
(no need to evaluate the $g_d$ integrals). At which dimensions
there is no solution with finite $T_c$?

[Note that the condensate is confined by the trap to a finite size
so that the system does not have a proper thermodynamic
($N\rightarrow \infty$) limit. Nonetheless, there is a reasonable
sharp crossover temperature $T_c$, at which a macroscopic fraction
of particles condenses to the ground state.]

\end{itemize}

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