28 From Classical Mechanics to Quantum Field Theory. A Tutorial
we will make use of the following identities which encode the orthonormality and
completeness properties of such a basis:
〈n|n′〉=δnn′, (1.126)
∑∞
n=0
|n〉〈n|=I. (1.127)
Acoherent stateis, by definition, an eigenstate|α〉of the annihilation opera-
tor^18 with eigenvalueα∈C:
a|α〉=α|α〉. (1.128)
An explicit expression for|α〉may be found by expanding it on the Fock basis:
|α〉=
∑∞
n=0an|n〉. After imposing (1.128), one finds:
|α〉=
∑∞
n=0
αn
√
n!
|n〉=
∑∞
n=0
αn
n!
(
a†
)n
| 0 〉=eαa
†
| 0 〉,∀α∈C. (1.129)
These states are not orthogonal, since:
〈α|β〉=eα
∗β
, (1.130)
but form a complete set, as:
I=
∫
(dReα)(dImα)
π
e−|α|
2
|α〉〈α|. (1.131)
Thus, the set: {|α ̃〉=e−|α|^2 /^2 eαa†| 0 〉}α∈Cis an overcomplete set of normalized
vectors.
We leave the details of the proof of these identities to the reader, by only notic-
ing that to show (1.130), one has to use the Baker-Campbell-Hausdorff formula
and the fact that:
eα
∗a
| 0 〉=
∑∞
n=0
(αa)n
n!
| 0 〉=| 0 〉.
To show (1.131), instead, one can make use of the following chain of identities:
∫ (dReα)(dImα)
π |α ̃〉〈α ̃|=
∑∞
n=0
∑∞
m=0
√^1
n!
√^1
m!
|n〉〈m|
∫ (dReα)(dImα)
π α
n(α∗)me−|α|^2
=
∑∞
n=0
∑∞
m=0
√^1
n!
√^1
m!
|n〉〈m|^1 π
∫ 2 π
0
dθ
∫∞
0
ρdρei(n−m)θρn+me−ρ
2
=
∑
n
∑
m
n!δnm√^1
n!
√^1
m!
|n〉〈m|=
∑∞
n=0
|n〉〈n|=I,
where polar coordinatesα=ρeiθhave been defined.
(^18) One can easily show that the creation operator does not admit eigenvectors with non-zero
eigenvalue.