Creation and annihilation operators

Creation operators and annihilation operators are <a href="/facts/Operator_(mathematics)/1VWgEdjS">mathematical operators</a> that have widespread applications in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, notably in the study of <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillators</a> and many-particle systems. An annihilation operator (usually denoted 
 
 
 
 
 
 
 a
 ^
 
 
 
 
 
 {\displaystyle {\hat {a}}}
 
) lowers the number of particles in a given state by one. A creation operator (usually denoted 
 
 
 
 
 
 
 
 a
 ^
 
 
 
 
 †
 
 
 
 
 {\displaystyle {\hat {a}}^{\dagger }}
 
) increases the number of particles in a given state by one, and it is the <a href="/facts/Hermitian_adjoint/gN6RHphd">adjoint</a> of the annihilation operator. In many subfields of <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Chemistry/SrNqkGVm">chemistry</a>, the use of these operators instead of <a href="/facts/Wavefunction/KgM5ykjY">wavefunctions</a> is known as <a href="/facts/Second_quantization/PPJHwX1Q">second quantization</a>. They were introduced by <a href="/facts/Paul_Dirac/6bt5yRdk">Paul Dirac</a>.
Creation and annihilation operators can act on states of various types of particles. For example, in <a href="/facts/Quantum_chemistry/H9AXV0lx">quantum chemistry</a> and <a href="/facts/Many-body_theory/uoMIVD3E">many-body theory</a> the creation and annihilation operators often act on <a href="/facts/Electron/L0KBo5cW">electron</a> states. They can also refer specifically to the <a href="/facts/Ladder_operators/kyGdlRjQ">ladder operators</a> for the <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillator</a>. In the latter case, the creation operator is interpreted as a raising operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent <a href="/facts/Phonons/tflWW6vw">phonons</a>. Constructing Hamiltonians using these operators has the advantage that the theory automatically satisfies the <a href="/facts/Cluster_decomposition_theorem/n820DGPp">cluster decomposition theorem</a>.
The mathematics for the creation and annihilation operators for <a href="/facts/Bosons/XtPFcjb5">bosons</a> is the same as for the <a href="/facts/Ladder_operators/kyGdlRjQ">ladder operators</a> of the <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillator</a>. For example, the <a href="/facts/Commutator/eYxzFmFY">commutator</a> of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for <a href="/facts/Fermions/LEckdDsR">fermions</a> the mathematics is different, involving <a href="/facts/Anticommutator/eYxzFmFY">anticommutators</a> instead of commutators.

Creation and annihilation operators open-in-new

Creation and annihilation operators