Consider a gas of $n$ identical spin-0 bosons confined by an isotropic three-dimensional harmonic oscillator potential. (In the rubidium experiment discussed above, the confining potential was actually harmonic, though not isotropic.) The energy levels in this potential are $\epsilon =nhf$, where $n$ is any nonnegative integer and $f$ is the classical oscillation frequency. The degeneracy of level $n\text{is}(n+1)(n+2)/2$.

(a) Find a formula for the density of states, $g\left(\epsilon \right)$, for an atom confined by this potential. (You may assume $n>>1$.)

(b) Find a formula for the condensation temperature of this system, in terms of the oscillation frequency $f$.

(c) This potential effectively confines particles inside a volume of roughly the cube of the oscillation amplitude. The oscillation amplitude, in turn, can be estimated by setting the particle's total energy (of order $kT$ ) equal to the potential energy of the "spring." Making these associations, and neglecting all factors of 2 and $\mathrm{\pi}$ and so on, show that your answer to part (b) is roughly equivalent to the formula derived in the text for the condensation temperature of bosons confined inside a box with rigid walls.