Finding the Prefactor in the Simple Harmonic Oscillator
Propagator
Michael Fowler 10/24/07
Hand-waving Argument
Recall that the free particle propagator has the form
(
)
(
)
2
,
;
,0
exp
.
2
2
im x
x
m
U
x T x
iT
T
π
⎧
⎫
′
−
⎪
⎪
′
=
⎨
⎬
⎪
⎪
⎩
⎭
=
=
From a classical mechanical evaluation of the action, we can show the simple harmonic
oscillator propagator has the form
(
)
(
)
(
)
2
2
,
;
,0
exp
cos
2
2 sin
im
U
x T x
A T
x
x
T
xx
T
ω
ω
ω
⎧
⎫
⎡
⎤
′
′
=
+
′
−
⎨
⎬
⎣
⎦
⎩
⎭
=
where
A
(
T
) is a so far unknown function of time.
For sufficiently small
T
, though, much less
than the period of the oscillator, the potential will have no significant effect, so the propagator
must tend to the free particle propagator. That is, for
T
tending to zero,
(
)
/ 2
.
A T
m
iT
π
=
=
But we also know that at time
2
/
t
π
ω
=
, all simple harmonic wavefunctions return to their
t
= 0
values, so a particle localized at
x
′
will be localized at
x
′
again after one period.
In the
exponent we do indeed see this cyclic behavior. In the prefactor, we must replace
T
by
(
)
sin
/
t
ω
ω
, that is,
( )
.
2
sin
m
A T
i
T
ω
π
ω
=
=
Note this expression is also consistent with the free particle propagator in the limit
ω
going to
zero, in other words, the vanishing of the simple harmonic oscillator potential.
Approximating Integrals by Stationary Phase Techniques
Actually, this result can be derived from the integral over the fluctuations about the classical
path.
The argument is closely analogous to that for the free particle, and the following equation
is a straightforward generalization of that case (discussed in the previous lecture):

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