Creation
Operator| Creation
Operator |
In General > s.a. annihilation [modified
versions]; fock
space [number
operator, generalizations]; Normal Order.
* Idea: An operator that
adds a quantum to a (free) field in quantum field theory, or raises the energy
of an oscillator by one level in quantum mechanics.
* In quantum mechanics:
The raising operator for the i-th degree of freedom of a system; Depends
on the choice of a parameter 
,
and can
be expressed as
ai = (i /2
)1/2 qi – i
(1/2i)1/2 pi ;
A choice of is equivalent to a choice of complex structure on phase
space.
* Properties: Bosonic
ones satisfy the commutation relations [ak
, ak
']
=
0, [ak, ak
']
= 
kk' and,
for general powers (with n 
m)
anan
= N (N–1) ··· (N–n+1)
, [an, am]
= 
k=1n k!
{m \choose k}{n \choose k} a(m–k) a(n–k)
;
Fermionic ones satisfy the commutation relations {bk, bk'}
=
kk'.
@ References: Szafraniec RPMP(07)
[characterizations]; Kim et al PRL(08)-a0901 [commutation
relations, for photons, proposed
experiment].
For Different Theories > s.a. Ladder
Operators; quantum
dirac fields.
* Harmonic oscillator:
One normally chooses = m
,
so H = (a a +
1/2); In the holomorphic representation,
a:= (1/
)
(
+ d/d)
, a:=
(1/)
( – d/d)
, where :=
(m/)1/2 x ;
Other relationships are that q = (/2m)1/2 (a + a); L3 =
i (a2 a1 – a1 a2).
* In quantum field theory:
The operator ak corresponding
to the coefficient of a negative-frequency mode in a field expansion
(x)
= k (ak uk(x)
+ a*k u*k(x))
;
In a Fock space 
(
):=
{
= (
0, 1, 2, 3,...)},
the creation operator a(
)
associated with any 
is
the adjoint of the corresponding annihilation operator a(),
a() :=
(0, 0 , 1 
, 
2 ,...)
; basically, a |n
=
(n+1)1/2 |n+1 .
@ References: Odake & Sasaki JMP(06)qp [solvable systems].
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jun 2009