A Short Course on Quantum Mechanics and Methods of Quantization 33
Given any|ψ〉∈H, we can use the resolution of the identity (1.131) to write:
|ψ〉=
(∫
(dRez)(dImz)
π e
−|z|^2 |z〉〈z|
)
|ψ〉=
∫
(dRez)(dImz)
π e
−|z|^2 |z〉ψ(z∗),
(1.151)
whereψ(z∗) ≡〈z|ψ〉is the wave functions associated to the vectorψ in the
coherent state basis.
It is not difficult to verify that, for any|ψ〉,|φ〉∈H:
〈ψ|φ〉=
∫
(dRez)(dImz)
π
e−|z|
2
ψ(z∗)∗φ(z∗), (1.152)
which shows that we are working in the Hilbert spaceHBFof all anti-holomorphic
functions inzsuch that:
‖ψ‖^2 BF≡
∫
(dRez)(dImz)
π
e−|z|
2
|ψ(z∗)|^2 <∞, (1.153)
thus obtaining the so-called Bargmann-Fock representation. This Hilbert space
might be thought of as the completion of the linear space of polynomials in the
variablez∗:P={P(z∗)=a 0 +a 1 z∗+···+an(z∗)n}with respect to the scalar
product defined by the measure:
dμ(z)≡
(dRez)(dImz)
π e
−|z|^2. (1.154)
In this representation, the vectors of the Fock basis are given by the monomials in
z∗. Indeed, from (1.129), one has:
Φn(z∗)=〈z|n〉=
∑∞
m=0
(z∗)m
√
m!
〈m|n〉=
(z∗)n
√
n!
. (1.155)
It is interesting to see how the creation/annihilation operatorsa†/aare repre-
sented onHBF. From the definition of a coherent state, we know thata|z〉=z|z〉,
i.e.:〈z|a†=z∗〈z|. Therefore we can write:
a†|z〉=
∑∞
m=0
(√z∗)m
m!
a†|m〉=
∑∞
m=0
(√z∗)m
m!
√
m+1|m+1〉
=
∑∞
n=1
(z√∗)n−^1
n!
n|n〉=
∂
∂z
∑∞
n=0
(√z∗)n
n!
|n〉.
This means thata†|z〉=∂z∂|z〉and〈z|a=∂z∂∗〈z|.Inotherwords,inHBF:
(i)aacts as the derivative with respect toz∗:
a:f(z∗)=〈z|f〉→〈z|a|f〉= ∂
∂z∗
〈z|f〉= ∂
∂z∗
f(z∗) ; (1.156)