Quantum Mechanics for Mathematicians

coordinatez(t) =√^12 (

√

ωq(t) +√iωp(t)) for the harmonic oscillator (see 22.1)

withz(0) =α 0.
Equations 23.3 and 23.9 can be used to calculate a propagator function in
terms of coherent states, with the result

〈αT|e−iHT|α 0 〉= exp(−

1

2

(|α 0 |^2 +|αT|^2 ) +αTα 0 e−iωT−

i

2

ωT) (23.10)

23.4.3 The position space propagator

Coherent states can be expressed in the position space representation by calcu-
lating

〈q|α〉=〈q|e−

|α|^2

(^2) eαa
†
| 0 〉
=e−
|α|^2
(^2) e
√α
2 (
√ωq−√ 1
ωdqd)(ω
π

)

(^14)
e−
ω 2 q 2

=(

ω

π

)

(^14)
e−
|α|^2
(^2) eα
√ω
2 qe−√α 2 ωdqde−α
2
(^4) e−ω 2 q
2

=(

ω

π

)

(^14)
e−
|α|^2
(^2) e−α
2
(^4) eα
√ω
2 qe−
ω 2 (q−√α
2 ω)
2

=(

ω

π

)

(^14)
e−
|α|^2
(^2) e−α
2
(^2) e−ω 2 q^2 e
√
2 ωαq (23.11)
This expression gives the transformation between the position space basis and
coherent state basis. The propagator in the position space basis can then be
calculated as
〈qT|e−iHT|q 0 〉=

1

π^2

∫

C^2

〈qT|αT〉〈αT|e−iHT|α 0 〉〈α 0 |q 0 〉d^2 αTd^2 α 0

using equations 23.10, 23.11 (and its complex conjugate), as well as equation
23.7.
We will not perform this (rather difficult) calculation here, but just quote
the result, which is

〈qT|e−iHT|q 0 〉=

√

ω

i 2 πsin(ωT)

exp

(

iω

2 sin(ωT)

((q^20 +qT^2 ) cos(ωT)− 2 q 0 qT)

)

(23.12)

One can easily see that asT→0 this will approach the free particle propagator
(equation 12.9, withm= 1)

〈qT|e−iHT|q 0 〉≈

√

1

i 2 πT

e

2 iT(qT−q^0 )^2

and as in that case becomes the distributionδ(q 0 −qT) asT →0. Without
too much difficulty, one can check that 23.12 satisfies the harmonic oscillator
Schr ̈odinger equation (inq=qTandt=T, for any initialψ(q 0 ,0)).
As in the free particle case, the harmonic oscillator propagator can be defined
first as a function of a complex variables=τ+iT, holomorphic forτ >0, then