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.