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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