PlanetPhysics/Pauli Exclusion Principle

The Pauli exclusion principle states that fermions are antisymmetric under particle exchange, and that as a consequence no two fermions may occupy the same quantum state. Mathematically, the exchange operator for a two-body wavefunction is $$ \hat{X} \psi(1, 2) = g \psi(2, 1) $$ Normalisation considerations tell us that the eigenvalue, $$g$$ must be either $$\pm 1$$ (as the operator must conserve probability). The Pauli exclusion principle then states that the eigenvalue is $$+1$$ for bosons and $$-1$$ for fermions, and that a wavefunction with an eigenvalue of $$-1$$ describes particles that cannot occupy the same quantum state. The spin-statistics theorem states that these particles are fermions, with half-integer spin.