From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 29

Coherent states satisfy several interesting properties that we now list:

(1) The mean value of creation/annihilation operators on a coherent state is

non-zero. More explicitly:

〈α ̃|a|α ̃〉=α,〈α ̃|a†|α ̃〉=α∗. (1.132)

(2) A coherent state is also denominated “displaced vacuum” since it can be

written as:

|α ̃〉=D(α)| 0 〉, (1.133)

wherewehaveintroducedthedisplacement operator

D(α)≡eαa

†−α∗a

, (1.134)

so called because:D†(α)aD(α)=a+α.

The latter expressions can be proved by noticing that:

D(α)=eαa

†−α∗a

=e

|α 2 |^2

eαa

†

e−α

∗a

,

D(α)†=e−αa

†+α∗a

=e−

|α 2 |^2

eα

∗a

e−αa

†

.

It is interesting to look at formula (1.133) in the coordinate representation, i.e.
when we chooseH=L^2 (R={q},dμ=dq) and:

qˆ=q ⇒ eıp^0 ˆqψ(q)=eıp^0 qψ(q), (1.135)

pˆ=−ı

d

dq

⇒ e−ıq^0 ˆpψ(q)=ψ(q−q 0 ). (1.136)

If we now recall that the vacuum | 0 〉is represented by the functionψ 0 (q)=
1
π^1 /^4 e

−q^2 / (^2) , with just some algebra we can show that|α ̃〉is represented by:
ψα(q)=eı
p 0 q 0
(^2) e−ıq^0 ˆpep^0 ˆqψ 0 (q)=^1
π^1 /^4
e−ı
p 0 q 0
(^2) eıp^0 qe−(q−q^0 )^2 /^2 , (1.137)
wherewehavesetα=(q 0 +ip 0 )/

√

2. This shows that, like the vacuum, a coher-
   ent state is represented by a wave function of gaussian type, with a mean value
   displaced fromq=0toq=q 0.
   The reader might have recognized in Eq. (1.133) the operators that define the
   Heisenberg-Weyl group that we have encountered in the previous section. Indeed,
   the set of coherent states that we have just defined gives an explicit example of an
   irreducible representation ofW 1.