= Capacity of an isotropic oscillator shell
{title2=$g_N^{\rm spatial}=(N+1)(N+2)/2$}
The number of nonnegative triples with $n_x+n_y+n_z=N$ is the stars-and-bars count $\binom{N+2}{2}$. Including a two-state <electron> <spin> gives capacity $(N+1)(N+2)$. Filling non-interacting shells therefore uses capacities $2,6,12,\ldots$, with shell <energy> $(N+3/2)\hbar\omega$ per <electron>. A partly filled shell has degenerate many-body <ground states>; rearranging its occupants is not a positive excitation <energy>.
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