A statistical description of a system of particles that obeys the rules of quantum mechanics rather than classical mechanics. In quantum statistics, energy states are considered to be quantized. Bose–Einstein statistics apply if any number of particles can occupy a given quantum state. Such particles are called bosons. Bosons have an angular momentum nh/2π‎, where n is zero or an integer and h is the Planck constant. For identical bosons the wave function is always symmetric. If only one particle may occupy each quantum state, Fermi–Dirac statistics apply and the particles are called fermions. Fermions have a total angular momentum (n+½)h and any wave function that involves identical fermions is always antisymmetric.

The relation between the spin and statistics of particles is given by the spin–statistics theorem. In two-space dimensions, it is possible that there are particles (or quasiparticles) that have statistics intermediate between bosons and fermions. These particles are known as anyons; for identical anyons the wave function is not symmetric (a phase sign of +1) or antisymmetric (a phase sign of −1), but interpolates continuously between +1 and −1. Anyons may be involved in the fractional quantum Hall effect.