Creation and annihilation operators

Creation operators add a quantum to a mode, while annihilation operators remove one. Their algebra encodes particle statistics and underlies the occupation-number description of quantum fields.

Occupation number: The number of particles occupying a specified quantum state or mode. Repeated creation and annihilation change this mode-by-mode particle count.
Quantum harmonic oscillator: A quantum system whose potential energy is proportional to displacement squared. Its ladder operators provide the prototype for creating and annihilating field quanta.
Second quantization: A framework that describes variable particle numbers using operators acting on Fock space. It uses creation and annihilation operators as its basic particle-changing operations.
First quantization: A quantum-mechanical framework that assigns wavefunctions to a fixed number of particles. Unlike this framework, creation and annihilation operators naturally change particle number.
Fock space: A Hilbert space built from states with varying particle numbers. Creation and annihilation operators connect its sectors with different particle counts.
Quantum field: An operator-valued field whose excitations describe particles and their interactions. Creation and annihilation operators arise when a field is expanded into modes.
Quantum electrodynamics: The quantum field theory of electromagnetic fields and charged particles. Photon creation and annihilation operators describe quantized electromagnetic modes.
Position operator: An operator representing the position observable of a quantum particle. It measures an observable; creation and annihilation operators change occupation.
Canonical commutation relations: Algebraic relations between canonical position and momentum operators, or bosonic field operators. For bosonic modes, these relations fix the algebra of creation and annihilation.
Boson: A particle with integer spin that obeys Bose–Einstein statistics. Bosonic mode operators commute, allowing unrestricted occupation of a mode.

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