It will be shown how to compute the density matrix for the harmonic oscillator:
using the functional integral representation. The density matrix is given by
As we saw in the last lecture, paths in the vicinity of the
classical path on the inverted potential give rise to the dominant
contribution to the functional integral. Thus, it proves useful to
expand the path 
about the classical path. We introduce
a change of path variables from to 
, where
where 
satisfies
subject to the conditions
so that 
.
Substituting this change of variables into the action integral yields
An integration by parts makes the cross terms vanish:
where the surface term vanishes becuase
and the second term vanishes because 
satisfies
the classical equation of motion.
The first term in the expression for S is the classical action, which we have seen is given by
Therefore, the density matrix for the harmonic oscillator becomes
where I[y] is the path integral
Note that I[y] does not depend on the points x and x' and therefore can only contribute an overall (temperature dependent) constant to the density matrix. This will affect the thermodynamics but not any averages of physical observables. Nevertheless, it is important to see how such a path integral is done.
In order to compute I[y] we note that it is a functional integral
over functions that vanish at 
and
. Thus, they are a special class of periodic
functions and can be expanded in a Fourier sine series:
where
Thus, we wish to change from an integral over the functions
to an integral over the Fourier expansion coefficients 
.
The two integrations should be equivalent, as the coefficients
uniquely determine the functions . Note that
Thus, terms in the action are:
Since the cosines are orthogonal between and ,
the integral becomes
Similarly,
The measure becomes
which, is not an equivalent measure (since it is not derived from a
determination of the Jacobian), but is chosen to give the correct
free-particle ( 
) limit, which can ultimately be corrected
by attaching an overall factor of 
.
With this change of variables, I[y] becomes
The infinite product can be written as
the product in the square brackets is just the infinite product
formula for 
, so
that I[y] is just
Finally, attaching the free-particle factor ,
the harmonic oscillator density matrix becomes:
Notice that in the free-particle limit 
,
and
, so that
which is the expected free-particle density matrix.