From Classical Mechanics to Quantum Field Theory

30 From Classical Mechanics to Quantum Field Theory. A Tutorial

1.3.1.2 Physical properties

Coherent states play an important role in physical problems. They were introduced
in the context of optics and possess some very interesting properties that we now
briefly present.

• Number of particle distribution.
  Coherent states are given by an infinite superposition of Fock states, hence
  they have no definite number of particles. However we may easily see that:

〈α ̃|N|α ̃〉=e−|α|

2 ∑

nm

(α∗)n

√

n!

(α)m

√

m!

〈n|N|m〉= e−|α|

2 ∑∞

n=0

|α|^2 n

n!

n

=e−|α|

2

|α|^2

∑∞

n=1

(

|α|^2

)n− 1

(n−1)!

=e−|α|

2

|α|^2

∑∞

m=0

|α|^2 m

m!

=|α|^2.

(1.138)

The last expression before the final result shows that the probability of

finding the valuemis given by:

pm=

|α|^2 m

m! e

−|α|^2. (1.139)

This means that, in a coherent state, the number of particles obeys a

Poisson distribution, with average|α|^2 , i.e. a distribution that could be

obtained form a classical counting of particles which are randomly dis-

tributed (with fixed mean).

• Quasi-classical states.
  Let us consider the equations of motion of a classical 1D oscillator:
  q ̇(t)=ωp(t), ̇p(t)=−ωq(t). Settingα(t)≡[q(t)+ıp(t)]/

√

2, we

can summarize them in the complex equation: ̇α(t)=−ıωα(t), whose

solution is easily found to be:

α(t)=α(t=0)e−ıωt≡α 0 e−ıωt, (1.140)

with energy

E=

ω

2

[

q(t)^2 +p(t)^2

]

=ω|α(t)|^2 =ω|α 0 |^2. (1.141)

We notice that formulas (1.116) give the quantum counterpart of what

we are doing here at the classical level. To compare the two cases, let

us recall that an operatorAˆevolves in time as: Aˆ(t)=eıtHAeˆ−ıtH so