3-D Harmonic Oscillator

Consider a particle subject to a central force F = -kr directed towards the origin and proportional to the distance away from the origin.  Then
U(r) = kr2 = mω2r2, with ω2 = k/m and F = - U(r).

The Hamiltonian is
H = P2/m + mω2R2 = (Px2 + Py2 + Pz2)/m + mω2(X2 + Y2 + Z2)
= Hx + Hy + Hz.
The state space E can be written as a tensor product space, E = Ex ⊗ Ey ⊗ Ez .
Hx acts in Ex , Hy acts in Ey , and Hz acts in Ez . We know the eigenfunctions if Hi in Ei.
Hini> = (ni + )ħω|Φni>.
{|Φni>} is an orthonormal basis for Ei.
{|Ψnx,ny,nz> =  |Φnx>⊗|Φny>⊗|Φnz>} is an orthonormal basis for E .
We have H|Ψn1,n2,n3> = (nx + ny + nz + 3/2)ħω|Ψnx,ny,nz>.
The energy levels of the three-dimensional harmonic oscillator are denoted by En = (nx + ny + nz + 3/2)ħω, with n a non-negative integer, n = nx + ny + nz
All energies except E0 are degenerate.  E0 = (3/2)ħω is not degenerate.


For the three-dimensional isotropic harmonic oscillator the energy eigenvalues are E = (n + 3/2)ħω, with n = n1 + n2 + n3, where n1, n2, n3 are the numbers of quanta associated with oscillations along the Cartesian axes.  Derive a formula for the degeneracy of the quantum state n, for spinless particles confined in this potential.