Consider a gas of N identical spin- 0 bosons confined by an isotropic three-dimensional harmonic oscillator potential. In experiments performed on rubidium, for example, the confining potential is harmonic, although not isotropic. The energy levels of this potential are given by ε=nhf, where f is the classical oscillation frequency and n is any nonnegative integer. This is because in three dimensions, the allowed energies of the isotropic simple harmonic oscillator are given by ε=(nx+ny+nz)hf. The degeneracy of each level n is given by (n+1)(n+2)/2 (you can verify this for n=1, for example).
a) Show that for n≫1, the density of states (the number of states per unit energy) is
p(ε)=ε²/2(hf)³