A Short Course on Quantum Mechanics and Methods of Quantization 31
that its mean value on a state,〈Aˆ〉t≡〈ψ|Aˆ(t)|ψ〉evolves according to the
equation:
ı
d
dt
〈Aˆ〉t=〈[A,ˆHˆ]〉. (1.142)
In particular, it is immediate to see that fora, a†andH=ω
(
a†a+1/ 2
)
one has:
〈a〉t=〈a〉 0 e−ıωt, 〈a†〉t=〈a†〉 0 eıωt, (1.143)
〈H〉t=ω
(
〈a†a〉t+^1
2
)
. (1.144)
These expectation values coincide with the classical solution (1.140) if:
〈a〉 0 =α 0 , 〈a†〉 0 =α∗ 0 (1.145)
and with the classical energy (1.141) if:
〈a†a〉 0 =|α 0 |^2 , (1.146)
up to a content term which becomes negligible in the (classical) limit
|α 0 |^2 >>ω.
A state that satisfies these conditions is called quasi-classical. It is easy
to see that a coherent state|α〉does indeed satisfy Eqs. (1.145, 1.146).
Moreover, it is not difficult to show the vice versa, i.e. that all quasi-
classical states are coherent states.
To show this result, we suppose that|ψ〉is a quasi-classical state and
consider the operatorb=a−α 0 I,sothat:
‖b|ψ〉‖^2 =〈ψ|b†b|ψ〉=〈ψ|
(
a†−α∗ 0
)
(a−α 0 )|ψ〉
=|α 0 |^2 −α 0 α∗ 0 −α∗ 0 α 0 +|α 0 |^2 =0.
Since the scalar product is non-degenerate, one must have
0=b|ψ〉=(a−α 0 )|ψ〉⇔a|ψ〉=α 0 |ψ〉,
showing that|ψ〉is the coherent state|α 0 〉.
- Minimal uncertainty states.
Coherent states are also called minimal uncertainty states because they
saturate Heisenberg uncertainty inequality:
ΔˆqΔˆp≥
2. (1.147)
This can be easily seen, by calculating:
〈α 0 |ˆq|α 0 〉=
α 0 +α∗ 0
√
2
, 〈α 0 |pˆ|α 0 〉=
α 0 −α∗ 0
√
2 i
