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.