Annihilation
Operator| Annihilation
Operator |
In General > s.a. creation
operator; fock
space [number operator, generalizations].
* Idea: An operator that removes a quantum from a (free) field, or
lowers the energy of an oscillator by one level.
* In quantum mechanics:
A lowering operator for the i-th
degree of freedom; Depends on the choice of value for 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.
* Harmonic oscillator:
One normally chooses =
m
, so H = (a
a +
1/2); In the holomorphic representation,
a := 2–1/2 (
+
d/d) , a :=
2–1/2 ( – d/d)
, where :=
(m/)1/2 x .
* In quantum field theory: The operator ak corresponding to the coefficient of a positive-frequency mode in a field expansion
= 
k(ak uk + a*k u*k)
;
In a Fock space 
(
):=
{
= (
0,
1, 2,
3,...)},
the annihilation operator a(
) associated with any 
is
a() :=
( · 1,
21/2 · 2,
31/2 · 3,
...) ; basically, a |n
= n1/2 |n–1 .
* Properties: Bosonic
ones satisfy the commutation relations [ak, ak ']
= 0 and [ak, ak ']
= 
kk
',
while fermionic ones satify {bk, bk '}
= kk
'; > s.a. creation
operator.
* On a Riemannian M: The Dirac operator d + on
the Hodge complex
of M.
References > s.a. Normal
Order.
@ General: Bueler mp/01 [on
Riemannian M];
Mizrahi & Dodonov JPA(02)qp [paradoxical
example]; Odake & Sasaki JMP(06)qp [solvable
systems].
@ 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]; > s.a. Ladder Operators.
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send feedback and suggestions to bombelli at olemiss.edu – modified
9 jun 2009