# Commutation Relations

## Commutation Relations

fundamental relations in quantum mechanics that establish the connection between successive operations on the wave function, or state vector, of two operators (*L̂ _{1} and L̂_{2}*) in opposite orders, that is, between

*L̂*and

_{1}L̂_{2}*L̂*. The commutation relations define the algebra of the operators. If the two operators commute, that is,

_{2}L̂_{1}*L̂*, then the corresponding physical quantities

_{1}L̂_{2}= L̂_{2}L̂_{1}*L*

_{1}and

*L*≤ ǀcǀ/2, where Δ

_{2}can have simultaneously defined values. But if their commutator is nonzero, that is,*L̂L̂*, then the uncertainty principle holds between the corresponding physical quantities: ΔL̂_{1}L̂_{2}– L̂_{2}L̂_{1}= c_{1}ΔL̂_{2}*L*

_{1}and Δ

*L*

_{2}are the uncertainties, or dispersions, of the measured values of the physical quantities

*L*

_{1}and

*L*

_{2}. The commutation relation between the operator of the generalized coordinate

*q̂*and its conjugate generalized momentum

*p̂*(

*q̂p̂*–

*p̂ĝ*=

*iħ*, where ħ is Planck’s constant) is very important in quantum mechanics. If the operator

*L̂*commutes with the operator of the total energy of the system (the Hamiltonian)

*Ĥ*, that is,

*L̂Ĥ*=

*ĤL̂*, then the physical quantity

*L*(its average value, dispersion, and so on) preserves its value in time.

The commutation relations for the operators of the creation *a ^{+}* and annihilation

*a*

^{–}of particles are of fundamental importance in the quantum mechanics of systems of identical particles and in quantum field theory. For a system of free (noninteracting) bosons, the particle creation operator in the state

*n*, and the annihilation operator for the same particle, , satisfy the commutation relation for fermions, the relation holds. The latter commutation relation is a formal expression of the Pauli principle.

V. B. BERESTETSKII