state spaceH=F, whereF is the space of holomorphic functions (satisfying
d
dzψ= 0) onCwith finite norm in the inner product
〈ψ 1 |ψ 2 〉=
1
π
∫
C
ψ 1 (z)ψ 2 (z)e−|z|
2
d^2 z (22.4)
whered^2 z=dRe(z)dIm(z). The spaceFis sometimes called “Fock space”. We
define the following two operators acting on this space:
a=
d
dz
, a†=z
Since
[a,a†]zn=
d
dz
(zzn)−z
d
dz
zn= (n+ 1−n)zn=zn
the commutator is the identity operator on polynomials
[a,a†] = 1
One finds
Theorem.The Bargmann-Fock representation has the following properties
- The elements
zn
√
n!
ofFforn= 0, 1 , 2 ,...are orthonormal. - The operatorsaanda†are adjoints with respect to the given inner product
onF. - The basis
zn
√
n!
ofFforn= 0, 1 , 2 ,...is complete.
Proof.The proofs of the above statements are not difficult, in outline they are
- For orthonormality one can compute the integrals
∫
C
zmzne−|z|
2
d^2 z
in polar coordinates.
- To show thatzanddzd are adjoint operators, use integration by parts.