From Classical Mechanics to Quantum Field Theory

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.