Past exam of the advanced quantum physics course of the University of Cambridge / 2013 / Paper 1 / 2 / a

For identical particles, the particle exchange operator exchanges both spatial and spin labels. The symmetrization postulate requires a symmetric full state for bosons and an antisymmetric full state for fermions. A product $\Psi =\varphi \chi$ must therefore have spatial and spin exchange eigenvalues whose product is $+1$ or $-1$, respectively; a general state may be a sum of such products.

For orthonormal spatial orbitals $u,v$, the combinations ${\varphi }_{\pm }=[u(1)v(2)\pm v(1)u(2)]/\sqrt{2}$ have opposite exchange parities. Two spin-one-half fermions combine ${\varphi }_{+}$ with the antisymmetric spin-one-half singlet state $(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$, or ${\varphi }_{-}$ with one of the three symmetric spin-one-half triplet states. Two particles in the same spatial orbital must be a spin singlet. This is the Pauli exclusion principle applied to complete spin orbitals, not a ban on sharing a spatial orbital with opposite spins.

A Slater determinant enforces fermionic antisymmetry for many occupied spin orbitals. Bosonic symmetrization instead uses a permanent; repeated orbital occupations are allowed. These exchange requirements apply even when particles do not interact.