Digression(Spin coherent states). One can perform a similar construction
replacing the groupH 3 by the groupSU(2), and the state| 0 〉by a highest weight
vector of an irreducible representation(πn,Vn=Cn+1)of spinn 2. Writing|n 2 〉
for such a highest weight vector, we have
π′n(S 3 )|n
2〉=
n
2|
n
2
〉, πn′(S+)|n
2〉= 0
and we can create a family of spin coherent states by acting on|n 2 〉by elements
ofSU(2). If we identify states in this family that differ only by a phase, the
states are parametrized by a sphere.
For the casen= 1, this is precisely the Bloch sphere construction of section
7.5, where we took as highest weight vector|^12 〉=
(
1
0
)
. In that case, all states
in the representation spaceC^2 were spin coherent states (identifying states that
differ only by scalar multiplication). For larger values ofn, only a subset of the
states inCn+1will be spin coherent states.
23.2 Coherent states and the Bargmann-Fock state space
One thing coherent states provide is an alternate complete set of norm one
vectors inH, so any state can be written in terms of them. However, these
states are not orthogonal (they are eigenvectors of a non-self-adjoint operator
so the spectral theorem for self-adjoint operators does not apply). The inner
product of two coherent states is
〈β|α〉=〈 0 |e−(^12) |β| 2
eβae−
(^12) |α| 2
eαa
†
| 0 〉
=e−
(^12) (|α| (^2) +|β| (^2) )
eβα〈 0 |eαa
†
eβa| 0 〉
=eβα−
(^12) (|α| (^2) +|β| (^2) )
(23.3)
and
|〈β|α〉|^2 =e−|α−β|
2
The Dirac formalism used for representing states as position space or momen-
tum space distributions with a continuous basis|q〉or|p〉can also be adapted
to the Bargmann-Fock case. In the position space case, with states functions of
q, the delta-function distributionδ(q−q′) provides an eigenvector|q′〉for the
Qoperator, with eigenvalueq′. As discussed in chapter 12, the position space
wavefunction of a state|ψ〉can be thought of as given by
ψ(q) =〈q|ψ〉
with
〈q|q′〉=δ(q−q′)