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Problem 3: Bose-Einstein condensation of diluted gases in harmonic traps

The Nobel Prize in Physics 2001 has been awarded for "the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates". Such experimental realizations of BEC rely on trapping cold atoms in a potential where laser cooling and evaporative cooling bring bosons to a temperature of the order of $\sim 10$ nK.

Close to its minimum, the potential can be expanded to second order, and has the form

$U(\mathbf {r} )={\frac {m}{2}}\sum _{\alpha }\omega _{\alpha }^{2}x_{\alpha }^{2}$

where we allow for the possibility of anisotropic confinement, with different frequencies along different directions.

(a) We are interested in the limit of wide traps such that $\hbar \omega \ll k_{B}T$, the the discreteness of the allowed energies can be largely ignored. Show that in the limit, the number of states $N(E)$ and the corresponding density of states $g(E)$ are given by