How this calculator works
For a shell n, the orbital angular momentum index l takes every integer from zero to n-1. Each l admits magnetic projections ml from -l to +l, giving 2l+1 spatial orbitals. Each orbital accommodates two electron spin states. Shell capacity is a count of allowed states, rather than a prediction that a particular atom fills shells in simple numerical order. If only l is known, n has a lower bound and no finite upper bound.
Formula
n = 1,2,...; l = 0,...,n-1; ml = -l,...,+l; ms = ±1/2. Shell orbitals = n²; shell capacity = 2n²; subshell capacity = 2(2l+1).
Worked example
The n = 2 shell contains an s subshell with one orbital and a p subshell with three orbitals: four orbitals and eight electron states in total. An l = 2 subshell has five orbitals and capacity ten.
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Frequently asked questions
Can a quantum number be fractional?
Electron spin projection can be ±1/2, but n, l and ml in this orbital model are integers.
Does l determine n uniquely?
No. It only requires n to be at least l+1.
Why is n capped at 20 here?
The cap keeps the displayed state table readable. The mathematical selection rules extend to higher n.