Subject: The Prefactor in the Simple Harmonic Oscillator Propagator
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Date: Tue, 12 Jul 2005 08:41:58 -0500

Finding the Prefactor in the Simple Harmonic Oscillator Propagator

Michael Fowler

Hand-waving Argument

Recall that the free particle propagator has the form

From a classical mechanical evaluation of the action, we can
show the simple harmonic oscillator propagator has the form

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,

But
we also know that at time t = 2p/w,
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 (sinw
t)/w,
that is,

Note this expression is also consistent with the free
particle propagator in the limit w
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, Shankar’s (8.6.14)

We see from this that including small deviations from the
path of stationary phase multiplies the leading exponential term by a prefactor
that does not depend on the classical path, only on the time taken.

To see how to evaluate this prefactor, it’s worth briefly
reviewing a stationary phase integral in ordinary space, that is,

where we assume that the real function f  has an absolute minimum at the point
(x10, … , xN0) and
take variables yi = xi –
xi0.
Then to leading order near the stationary point, f(x) =
f(x0) + ½ yi Aij
yj.  Taking
only this term, the integral becomes

The matrix A
is positive definite and symmetric, it has real orthogonal eigenvectors, and
we can choose our coordinate axes in the directions of those
eigenvectors.  In this case,
A is a diagonal matrix, and its diagonal elements are just its
eigenvalues li.

So I  becomes:

Recall that Det A is invariant under an orthogonal
transformation, and so is unaffected by transforming back from the eigenvector
basis to the original coordinates.

Applying Stationary Phase to the Integral Over Paths

We’re now ready to generalize this result from an integral
over an N-dimensional space of yi’s to an integral over
the infinite dimensional space of paths y(t¢¢)
with boundary conditions y(0) = 0 and y(t) = 0.  Note that for the particular case at
hand, the Simple Harmonic Oscillator, the leading order quadratic term is in
fact the whole story, and so gives the exact result. This is not, of
course, true for general quantum systems.

As a preliminary step, we integrate by parts to write

The path
integral then becomes:

Comparing this with the finite-dimensional version, we see
that the N component vector {yi} is replaced by the
continuous function y(t¢¢),
and the matrix A by a differential operator acting on the space of path
functions y(t) equal to zero at t¢¢
= 0 and t¢¢
= t.

The operator is well-defined and Hermitian, with eigenstates
and eigenvalues:

Any reasonable path satisfying the boundary conditions can be
written as a sum over these eigenstates with Fourier coefficients
yn,

The functional integral over all paths now becomes an
infinite product of ordinary integrals over the variables
yn.  This is
precisely analogous to the finite integral above, except that there is no
maximum value N. The operator is diagonal with respect to this base
formed of its eigenstates, and therefore can be represented as an infinite
diagonal matrix with the diagonal elements equal to the eigenvalues—and, just as
for the finite matrix case, the gaussian integrals can be carried out in
succession, each one giving a term

When we take the infinite product of all these terms, we find
in the denominator the square root of the product of all the eigenvalues—and
that product is the determinant of the operator.  That is to say,

In the last step, we used the infinite-product representation
for the sine function:

The rather disconcerting factor K(t), which
does not depend on w
and hence has nothing to do with the dynamics, can be thrown in with similar
time-dependent and constant factors involved in the measure of the integral over
paths, and their overall contribution can be nailed down by the simple
observation that for w
= 0, we must recover the known free-particle propagator.  So, we can be quite careless about
overall multiplying factors!

Therefore, matching to the free-particle propagator in the
limit w
= 0, we findfor the simple harmonic oscillator.

I used an excellent book, Solitons and Instantons, by R. Rajaraman, for
this section. The original work was all by Feynman.