The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Commutation Relations

fundamental relations in quantum mechanics that establish the connection between successive operations on the wave function, or state vector, of two operators (L̂₁ and L̂₂) in opposite orders, that is, between L̂₁ L̂₂ and L̂₂ L̂₁. The commutation relations define the algebra of the operators. If the two operators commute, that is, L̂₁ L̂₂ = L̂₂ L̂₁, then the corresponding physical quantities L₁ and L₂ can have simultaneously defined values. But if their commutator is nonzero, that is, L̂L̂₁ L̂₂ – L̂₂ L̂₁ = c, then the uncertainty principle holds between the corresponding physical quantities: ΔL̂₁ΔL̂₂ ≤ ǀcǀ/2, where ΔL₁ and ΔL₂ are the uncertainties, or dispersions, of the measured values of the physical quantities L₁ and L₂. 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