Annihilation Operator
From Qwiki
In Quantum Mechanics one often uses the canonical commutation relation:
![[a,a^\dagger]=1](/images/math/4/4/d/44dc68d4be9a19778b6c295f03c85c88.png)
The creation (
) and annihilation (a) operators are generally used to describe processes in which bosons are created or destroyed. When 
obey the the anti-commutation relation 
the operators act on fermion fields. The boson creation and annihilation operators are directly analogous to the raising and lowering ladder operators that are used to describe the quantum mechanical simple harmonic oscillator. It is important to note, however, that there is a fundamental difference between the two cases. The SHO describes the behavior of a single particle living in a single-particle Hilbert space. The general creation and annihilation operators act in a multi-particle Fock Space.
The creation and annihilation operators act on number states in the Fock Space by adding or subtracting from the number of excitations:
The creation and annihilation operators are a very general and apply to a wide variety of settings. They frequently appear in quantum optics when the Electro-Magnetic Field is quantized.
Common Applications
Photon Counting
The combination 
is commonly referred to as the number operator. It acts on a number state 
as
That is to say, an eigenstate of the number operator is a state of definite particle number.
A coherent state 
is defined as an eigenstate of the annihilation operator,
and defines a classical state of the field with mean photon number | α | 2.
References
additional references TBA
- D. Suter, The Physics of Laser-Atom Interactions, Cambridge University Press, Cambridge, UK (1997) (ISBN 0-521-46239-8).
