From Classical Mechanics to Quantum Field Theory

32 From Classical Mechanics to Quantum Field Theory. A Tutorial

and

〈α 0 |qˆ^2 |α 0 〉=

1+(α 0 +α∗ 0 )^2

2 

, 〈α 0 |pˆ^2 |α 0 〉

1 −(α 0 −α∗ 0 )^2

2 

so that

Δˆq=

√

〈qˆ^2 〉−〈qˆ〉^2 =

√



2

,

Δˆp=

√

〈pˆ^2 〉−〈ˆp〉^2 =

√



2.

Let us notice that, for a coherent state, both Δˆq and Δˆpare minimal, and
equal to

√

/2. It is possible to define also the so-called “squeezed” states for
which:

Δˆq=√^1

2

e−s and Δˆp=√^1

2

e+s, (1.148)

withs∈R, which are again of minimal uncertainty. They can be constructed by
applying the displacement operator after having applied to the vacuum another
operator, called squeezing operatorδ(s):

|ψs〉=D(α)δ(s)| 0 〉 with δ(s)≡e

s 2 (a (^2) −a† (^2) )

. (1.149)

In the coordinate representation, a squeezed state is given by the wave-function:

ψs(q)=π−

(^14)
e
s 2
exp

(

ı

q 0 p 0

2

)

exp

[

−

(q−q 0 )^2

2 e^2 s

]

, (1.150)

i.e. by a Gaussian distribution centered inq=q 0 and variancees. Such states
are usually obtained in non-linear interaction problems in optics, when one adds
to the harmonic oscillator Hamiltonian a term of the kind:Hint=a^2 +(a†)^2.
We can summarize what we have seen in this subsection by saying that coherent
states are those states (and the only ones) for which the quantum expectation
values of the observables satisfy the same dynamical laws as the corresponding
classical functions (position, momentum, energy) on phase space and for which
the corresponding variances about such classical values get minimized.

1.3.1.3 Bargmann-Fock representation

In this subsection, instead of using greek lettersα, β,···, we will denote variables
inCwithz,z′,···and therefore a coherent states will be represented by the ket
|z〉and its corresponding eigenvalue with respect to the operatorais denoted with
z∈C.