= Spin orbital
{title2=$u_{a\sigma}(\mathbf r,s)=\phi_a(\mathbf r)\chi_\sigma(s)$}
When spatial and <spin> factors separate, a <spin orbital> combines a <spatial orbital> $\phi_a$ with a normalized <spin> state $\chi_\sigma$. For an <Electron>, choose <spin one-half along an axis> with $\sigma=\uparrow,\downarrow$. The two <spin orbitals> $u_{a\uparrow}$ and $u_{a\downarrow}$ are <orthogonal> even though their <spatial orbital> is identical, since their <inner product> contains $\langle\chi_\uparrow|\chi_\downarrow\rangle=0$. When spatial and <spin> variables do not separate, the complete <one-particle state> is instead a multicomponent <wavefunction> $u(\mathbf r,s)$; exclusion still applies to that complete <quantum state>.
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