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 ϵ=nhf, where n is any nonnegative integer and f is the classical oscillation frequency. The degeneracy of level n is (n+1)(n+2)/2.
(a) Find a formula for the density of states, g(ϵ), 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 π 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.