Creation Operator |
In Quantum Mechanics
> s.a. annihilation operator [including modified versions];
Ladder Operators; Normal Order;
Raising Operator.
* Idea: An operator that
takes a state in a family labeled by a discrete parameter to a higher one;
E.g., it raises the energy of an oscillator by one level.
$ Def: The raising operator
for the i-th degree of freedom for a system, which depends on the
choice of value for a parameter τ, and can be expressed as
a†i = (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:= (1/\(\sqrt2\)) (ξ + d/dξ) , a†:= (1/\(\sqrt2\)) (ξ − d/dξ) , where ξ:= (mω/\(\hbar\))1/2 x ;
Other relationships are that q = (\(\hbar\)/2mω)1/2
(a + a†); L3 =
i\(\hbar\) (a2† a1
− a1† a2).
> Specific theories:
see relativistic quantum particle [3D, deformed algebra].
In Quantum Field Theory > s.a. approaches to quantum
field theory [covariant]; fock space [number operator, generalizations].
* Idea: An operator that adds
a quantum of given momentum k to a (free) field.
* In quantum field theory:
The operator ak†
corresponding to the coefficient of a negative-frequency mode in a field expansion
φ(x) = ∑k (ak uk(x) + a*k u*k(x)) ;
In a Fock space \(\cal F\)(\(\cal H\)):= {ψ = (α0, α1, α2, α3, ...)}, the creation operator a†(σ) associated with any σ ∈ \(\cal H\) is the adjoint of the corresponding annihilation operator a(σ),
a†(σ) ψ:= (0, α0 σ, \(\sqrt2\) α1 ⊗ σ, \(\sqrt3\) α2 ⊗ σ, ...) ; basically, a† |n\(\rangle\) = (n+1)1/2 |n+1\(\rangle\) .
* Properties: Bosonic ones satisfy the commutation relations [ak†, ak'†] = 0, [ak , ak'†] = δkk' and, for general powers (with n ≤ m)
a†n an = N (N−1) ··· (N−n+1) , [an, a†m] = ∑k=1n k! \({m \choose k}{n \choose k}\) a†(m−k) a(n−k) ;
* Different commutation relations: Note that composite bosons
satisfy non-standard commutation relations (> see particle statistics),
and fermionic operators satisfy the anticommutation relations {bk,
bk'†}
= δkk'.
@ References:
Szafraniec RPMP(07) [characterizations];
Kim et al PRL(08)-a0901 [commutation relations, for photons, proposed experiment];
Gupta & Kumar a1105
[canonical brackets from continuous symmetries, BRST formalism];
Guadagnini JPA(13)-a1212 [representation];
Kumar et al PRL(13)
+ Zavatta & Bellini Phy(13)
[characterization by quantum process tomography].
> Specific theories:
see quantum dirac fields.
References > s.a. Normal Order.
@ References: Odake & Sasaki JMP(06)qp [solvable systems].
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