Quantum Mechanics for Mathematicians

taking the boundary value asτ→0. This fixes the branch of the square root in
23.12 and one finds (see for instance section 7.6.7 of [107]) that the square root
factor needs to be taken to be
√
ω
i 2 πsin(ωT)

=e−i

π 4

e−in

π 2

√

ω

2 π|sin(ωT)|

forωT∈[nπ,(n+ 1)π].

23.5 The Bargmann transform

The Stone von-Neumann theorem implies the existence of:

Definition.Bargmann transform
There is a unitary map called the Bargmann transform

B:HS→F

intertwining the Schr ̈odinger representation and the Bargmann-Fock represen-
tation, i.e., with operators satisfying the relation

Γ′S(X) =B−^1 Γ′BF(X)B

forX∈h 3.

In practice, knowingBexplicitly is often not needed, since the representation
independent relation

a=

1

√

2

(Q+iP)

can be used to express operators either purely in terms ofaanda†, which have
a simple expression

a=

d

dz

, a†=z

in the Bargmann-Fock representation, or purely in terms ofQandPwhich have
a simple expression

Q=q, P=−i

d

dq

in the Schr ̈odinger representation.
To compute the Bargmann transform one uses equation 23.11, for non-
normalizable continuous basis states|δu〉, to get

〈q|δu〉=

(ω

π

)^14

e−

u 22

e−

ω 2 q 2

e

√

2 ωuq

and

〈δu|q〉=

(ω

π

)^14

e−

u 22

e−

ω 2 q 2

e

√

2 ωuq (23.13)