Annihilation Operator |
In Quantum Mechanics > s.a. creation operator;
Lowering Operator; Normal Order.
* Idea: An operator that takes
a state in a family labeled by a discrete parameter to a lower one; E.g., it
lowers the energy of an oscillator by one level.
$ Def: A lowering operator for
the i-th degree of freedom, which depends on the choice of value for
a parameter τi, and
can be expressed as
ai = (τi /2\(\hbar\))1/2 qi + i (1/2\(\hbar\)τi)1/2 pi ;
A choice of value for τ is equivalent to a choice of complex structure on phase space.
* Harmonic oscillator: One normally chooses τ
= mω, so H = \(\hbar\)ω (a†a
+ 1/2); In the holomorphic representation,
a := 2−1/2 (ξ + d/dξ) , a† := 2−1/2 (ξ − d/dξ) , where ξ:= (mω/\(\hbar\))1/2 x .
In Quantum Field Theory > s.a. approaches to quantum
field theory [covariant]; fock space [number operator, generalizations].
* Idea: An operator that removes a quantum
of given momentum k from a (free) field.
$ Def: The operator
ak corresponding
to the coefficient of a positive-frequency mode in a field expansion
φ = ∑k (ak uk + a*k u*k) ;
In a Fock space \(\cal F\)(\(\cal H\)):= {ψ = (α0, α1, α2, α3,...)}, the annihilation operator a(σ) associated with any σ ∈ \(\cal H\) is
a(σ) ψ:= (σ · α1, 21/2 σ · α2, 31/2 σ · α3, ...) ; basically, a |n\(\rangle\) = n1/2 |n−1\(\rangle\) .
* Properties: Bosonic ones satisfy
the commutation relations [ak,
ak ']
= 0 and [ak,
ak '†]
= δkk',
while fermionic ones satify {bk,
bk '†}
= δkk' (Note that composite bosons satisfy
non-standard commutation relations, > see particle statistics);
> s.a. creation operator.
* On a Riemannian M: The Dirac operator
d + δ on the Hodge complex of M.
References > s.a. Normal Order.
@ General: Bueler mp/01 [on Riemannian manifolds];
Mizrahi & Dodonov JPA(02)qp [paradoxical example];
Odake & Sasaki JMP(06)qp [solvable systems];
Guadagnini JPA(13)-a1212 [representation].
@ Fermionic: Derrick JMP(63)
[representation on the space of periodic functions on a real interval].
@ Generalized / modified: Ghosh JMP(98);
Bagarello JMP(07)-a0903 [bounded version];
Trifonov JPA(12) [non-linear, and coherent states];
> s.a. Ladder Operators; relativistic
quantum particle [3D, deformed algebra].
@ Related topics:
Petrović a1001 [analytic functions];
Gupta & Kumar a1105
[canonical brackets from continuous symmetries, BRST formalism];
Kumar et al PRL(13)
+ Zavatta & Bellini Phy(13)
[bosonic, experimental characterization by quantum process tomography].
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