In quantum mechanics, the exchange operator
The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state
In three or higher dimensions, the exchange operator can represent a literal exchange of the positions of the pair of particles by motion of the particles in an adiabatic process, with all other particles held fixed.
Consider two repeated operations of such a particle exchange: Therefore,
is not only unitary but also an operator square root of 1, which leaves the possibilities Both signs are realized in nature.
Particles satisfying the case of +1 are called bosons, and particles satisfying the case of −1 are called fermions.
The spin–statistics theorem dictates that all particles with integer spin are bosons whereas all particles with half-integer spin are fermions.
The exchange operator commutes with the Hamiltonian and is therefore a conserved quantity.
Therefore, it is always possible and usually most convenient to choose a basis in which the states are eigenstates of the exchange operator.
Such a state is either completely symmetric under exchange of all identical bosons or completely antisymmetric under exchange of all identical fermions of the system.
In 2 dimensions, the adiabatic exchange of particles is not necessarily possible.
Instead, the eigenvalues of the exchange operator may be complex phase factors (in which case
The exchange operator is not well defined in a strictly 1-dimensional system, though there are constructions of 1-dimensional networks that behave as effective 2-dimensional systems.
In this method, one often defines an energetic exchange operator as: where
are the one-electron wavefunctions acted upon by the exchange operator as functions of the electron positions, and
[3] The labels 1 and 2 are only for a notational convenience, since physically there is no way to keep track of "which electron is which".