= Counting spin orbitals in an electron shell
{title2=$N_{\max}(n)=2n^2$}
Choose the <one-particle states> of a <hydrogen atom>, or a central-field description of a <chemical atom>, with labels
$$
n=1,2,\ldots,\qquad \ell=0,1,\ldots,n-1,\qquad m_\ell=-\ell,\ldots,\ell,\qquad m_s=\pm\tfrac12.
$$
The <principal quantum number> $n$ identifies the <electron shell>. The <orbital angular momentum> label $\ell$ identifies the <electron subshell>; the <magnetic quantum number> $m_\ell$ labels $2\ell+1$ independent <spatial orbitals>. Each has two <spin one-half along an axis> choices, hence $2(2\ell+1)$ <spin orbitals>. Applying the <Pauli exclusion principle> once per <spin orbital> gives
$$
N_{\max}(n)=\sum_{\ell=0}^{n-1}2(2\ell+1)=2\left(2\frac{n(n-1)}2+n\right)=2n^2.
$$
These <quantum numbers> label <wavefunctions> and <spin> projections, not <Electron> trajectories. They need not resolve exact <energy eigenstates> in the presence of <spin-orbit interaction>; changing to coupled <angular momentum eigenstates> preserves the count of independent <one-particle states> within this shell model.
\Image[/pauli-shell-capacity-counting.png]
{title=Counting complete spin orbitals gives electron-shell capacities $2,8,18,32$}
{description=Original counting diagram. The left bars add <electron subshell> capacities $2,6,10,14$; the right panel resolves the $n=3$ <electron shell> into eighteen <spin orbitals>. Each small cell denotes a label choice, not a region of space.}
For example, $n=3$ allows $\ell=0,1,2$, conventionally $3s,3p,3d$. Their <spatial orbital> counts are $1,3,5$; their <spin orbital> counts are $2,6,10$. The sum is eighteen. The algebra behind “one per complete state” is short: <two-particle antisymmetrization> gives $u\wedge u=(u\otimes u-u\otimes u)/\sqrt2=0$, and the zero <vector> cannot be a normalized <quantum state>. Opposite-<spin> <Electrons> in the same <spatial orbital> have distinct complete states, so their antisymmetric product survives.
This supplies a structural link to the <periodic table>: finite <electron subshell> capacities make repeated patterns of occupied <electron configurations> possible. It does not predict the exact <ground state> <electron configuration> or <energy> of a many-<Electron> <chemical atom>. Different <electron subshells> overlap in <energy>, and <Electron> interactions change their ordering. In particular, the <periodic table period> lengths are not simply successive $2n^2$ values: the third <periodic table period> has eight <chemical elements>, although the full $n=3$ <electron shell> has eighteen available states. A capacity counts available <spin orbitals>; an actual filling order requires a <Quantum Hamiltonian>.
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