= Solution
<electron spin> is intrinsic <angular momentum>, independent of an orbital <wavefunction>'s $\mathbf r\times\mathbf p$. It has $s=1/2$, so
$$
\boxed{S^2=\frac34\hbar^2,\qquad S_z=\pm\frac\hbar2,\qquad
S_i=\frac\hbar2\sigma_i,\qquad[S_i,S_j]=i\hbar\epsilon_{ijk}S_k.}
$$
A spatial orbital therefore carries a two-component <spinor> degree of freedom. Noncommuting components do not describe an ordinary <vector> with simultaneously definite projections; measurements along one axis give only the two allowed values. Even a spherically symmetric $l=0$ orbital retains <electron spin>.
The associated magnetic moment is $\boldsymbol\mu=-g_e e\mathbf S/(2m_e)$ with $e>0$ and $g_e\simeq2$. A <spinor> changes sign under a $2\pi$ <rotation> and returns under $4\pi$; that sign is observable only relative to another coherent <amplitude>, not as an isolated <global phase>. A literal classically rotating charged sphere does not account for this fixed half-integer <representation>, component algebra and <spinor> <rotation> law. Classical <angular momentum> provides useful analogies for torque and precession, but no ordinary classical orientation reproduces the full intrinsic quantum <spin> structure.
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