Anisotropic quantum harmonic oscillator ($E_{n_x{n}_y{n}_z}=\hslash {\sum }_i{\omega }_i(n_i+1/2)$)

For a positive diagonal quadratic potential $V=m{\sum }_i{\omega }_i^2{x}_i^2/2$, coordinate Hamiltonians commute and wavefunctions separate into one-dimensional oscillator factors. Equality of two squared frequencies gives rotation symmetry in their plane; equality of all three gives full rotation symmetry. Equalities create energy degeneracies and extra conserved angular momentum. Merely finding an accidental equality between particular level energies does not establish a global rotational symmetry.

The number of nonnegative triples with $n_x+n_y+n_z=N$ is the stars-and-bars count $\left(\genfrac{}{}{0ex}{}{N+2}{2}\right)$. Including a two-state electron spin gives capacity $(N+1)(N+2)$. Filling non-interacting shells therefore uses capacities $2,6,12,\dots$, with shell energy $(N+3/2)\hslash \omega$ per electron. A partly filled shell has degenerate many-body ground states; rearranging its occupants is not a positive excitation energy.