[[second quantization]] Creation and annihilation operators are pairs [anti-linear](Anti-linear%20map.md) operators where we typically define the _creation operator_ as $\hat{a}^\dagger$ and the _annihilation operator_ $\hat{a}$ that, as their name suggests, when applied to a [fock state](fock%20state.md), lead to the creation or annihilation of particle - or more generally a quanta. Locally, _creation_ and _annihilation_ model _emission_ and _absorption._ Additional properties of whatever particle is created or annihilated are generally denoted in the subscripts of the operators. Since [creation and annihilation operators](creation%20and%20annihilation%20operators.md) increase or decrease the eigenvalues of [number operators](number%20operator.md) respectively on [fock states](fock%20state.md) they are a type of [ladder operator.](Ladder%20operators.md) # Properties These are operators that are neither Hermitian, nor unitary, nor linear. We list their mathematical properties below. # Bosonic creation and annihilation operators The [Bosonic](Boson.md) creation and annihilation operators are mathematically equivalent to [Harmonic oscillator ladder operators](Harmonic%20Oscillator%20Ladder%20Operators.md) and are built on the same [commutation relation.](Commutators%20in%20quantum%20mechanics.md) # Fermionic creation and annihilation operators The algebraic properties of [Fermionic](Fermion.md) creation and annihilation operators are built on [anti-commutation](Anti-commutators%20in%20quantum%20mechanics.md) relations. ^de3105 # Generalized creation and annihilation operators #QuantumMechanics/StationaryStateQuantumSystems #QuantumMechanics/QuantumDynamics