Quantum Mechanics for Mathematicians

Definition(Coherent states).The coherent states inHare the states

|α〉=D(α)| 0 〉=eαa

†−αa

| 0 〉

whereα∈C.

Using the Baker-Campbell-Hausdorff formula

eαa

†−αa

=eαa

†

e−αae−

|α|^2

(^2) =e−αaeαa
†
e
|α|^2
2
so
|α〉=eαa
†
e−αae−
|α 2 |^2
| 0 〉
and sincea| 0 〉= 0 this becomes
|α〉=e−
|α|^2
(^2) eαa†| 0 〉=e−
|α|^2
2

∑∞

n=0

αn

√

n!

|n〉 (23.2)

Sincea|n〉=

√

n|n− 1 〉

a|α〉=e−

|α|^2

2

∑∞

n=1

αn

√

(n−1)!

|n− 1 〉=α|α〉

and this property could be used as an equivalent definition of coherent states.
In a coherent state the expectation value ofais

〈α|a|α〉=〈α|

1

√

2

(Q+iP)|α〉=α

so
〈α|Q|α〉=

√

2Re(α), 〈α|P|α〉=

√

2Im(α)
Note that coherent states are superpositions of different states|n〉, so are
not eigenvectors of the number operatorN, and do not describe states with a
fixed (or even finite) number of quanta. They are eigenvectors of

a=

1

√

2

(Q+iP)

with eigenvalueαso one can try and think of

√

2 αas a complex number whose
real part gives the position and imaginary part the momentum. This does not
lead to a violation of the Heisenberg uncertainty principle since this is not a
self-adjoint operator, and thus not an observable. Such states are however very
useful for describing certain sorts of physical phenomena, for instance the state
of a laser beam, where (for each momentum component of the electromagnetic
field) one does not have a definite number of photons, but does have a definite
amplitude and phase.