Quantum Mechanics for Mathematicians

In the Bargmann-Fock case, there is an analog of the distributional states
|q〉, given by taking states that are eigenvectors fora, but unlike the|α〉, are
not normalizable. We define

|δw〉=ewa

†

| 0 〉=

∑∞

n=0

wn

√

n!

|n〉=ewz=e

|w|^2

(^2) |w〉
Instead of equation 23.3, such states satisfy
〈δw 1 |δw 2 〉=ew^1 w^2
The|δw〉behave in a manner analogous to the delta-function, since the Bargmann-
Fock analog of computing〈q|ψ〉using the function space inner product is, writ-
ing
ψ(w) =

∑∞

n=0

cn

wn

√

n!

the computation

〈δz|ψ〉=

1

π

∫

C

ezwe−|w|

2

ψ(w)d^2 w

=

1

π

∫

C

ezwe−|w|

2 ∑∞

n=0

cn

wn

√

n!

d^2 w

=

1

π

∫

C

∑∞

m=0

zm

√

m!

wm

√

m!

e−|w|

2 ∑∞

n=0

cn

wn

√

n!

d^2 w

=

∑∞

n=0

cn

zn

√

n!

=ψ(z)

Here we have used the orthogonality relations
∫

C

wmwne−|w|

2

d^2 w=πn!δn,m (23.4)

It is easily seen that the Bargmann-Fock wavefunction of a coherent state is
given by

〈δz|α〉=e−

|α|^2

(^2) eαz (23.5)
while for number operator eigenvector states
〈δz|n〉=
zn
√
n!
In section 23.5 we will compute the Bargmann-Fock wavefunction〈δz|q〉for
position eigenstates, see equation 23.13.
Like the|α〉(and unlike the|q〉or|p〉), these states|δw〉are not orthogonal
for different eigenvalues ofa, but they span the state space, providing an over-
complete basis, and satisfy the resolution of the identity relation
1 =

1

π

∫

C

|δw〉〈δw|e−|w|

2

d^2 w (23.6)