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

$\displaystyle \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.



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