The Bargmann transform is then given by
(Bψ)(z) =
∫+∞
−∞
〈δz|q〉〈q|ψ〉dq
=
(ω
π
) (^14)
e−
z^2
2
∫+∞
−∞
e−
ω
2 q
2
e
√
2 ωzqψ(q)dq (23.14)
(hereψ(q) is the position space wavefunction) while the inverse Bargmann trans-
form is given by
(B−^1 φ)(q) =
∫
C
〈q|δu〉〈δu|φ〉e−|u|
2
d^2 u
=
(ω
π
)^14
e−
ω 2 q^2
∫
C
e−
u 22
e
√ 2 ωuq
φ(u)e−|u|
2
d^2 u
(hereφ(z) is the Bargmann-Fock wavefunction).
As a check of equation 23.14, consider the case of the lowest energy state in
the Schr ̈odinger representation, where| 0 〉has coordinate space representation
ψ(q) = (
ω
π
)
(^14)
e−
ωq^2
2
and
(Bψ)(z) =
(ω
π
) 14 (ω
π
) (^14)
e−
z 22
∫+∞
−∞
e−
ω 2 q^2
e
√
2 ωzqe−ωq (^22) dq
=
(ω
π
)^12 ∫+∞
−∞
e−
z 22
e−ωq
2
e
√
2 ωzqdq
=
(ω
π
)^12 ∫+∞
−∞
e
−ω
(
q−√z 2 ω
) 2
dq
=1
which is the expression for the state| 0 〉in the Bargmann-Fock representation.
For an alternate way to compute the harmonic oscillator propagator, the
kernel corresponding to applying the Bargmann transform, then the time evo-
lution operator, then the inverse Bargmann transform can be calculated. This
will give
〈qT|e−iHT|q 0 〉=
∫
C
〈qT|δu〉〈δu|e−iTH|q 0 〉e−|u|
2
d^2 u
=e−i
ωT 2
∫
C
〈qT|δu〉〈δue−iωT|q 0 〉e−|u|
2
d^2 u
from which 23.12 can be derived by a (difficult) manipulation of Gaussian inte-
grals.