The density matrix for the free particle
will be calculated by doing the discrete path integral
explicitly and taking the limit 
at
the end.
The density matrix expression is
Let us make a change of variables to
The inverse of this transformation can be worked out explicitly, giving
The Jacobian of the transformation is simply
Let us see what the effect of this transformation is for the case P=3. For P=3, one must evaluate
According to the inverse formula,
Thus, the sum of squares becomes
From this simple exmple, the general formula can be deduced:
Thus, substituting this transformation into the integral gives
where
and the overall prefactor has been written as
Now each of the integrals over the u variables can be integrated over independently, yielding the final result
In order to make connection with classical statistical mechanics, we
note that the prefactor is just 
, where 
is the kinetic prefactor that showed up also in the classical
free particle case. In terms of , the free particle
density matrix can be written as
Thus, we see that represents the spatial width of
a free particle at finite temperature, and is called
the ``thermal de Broglie wavelength.''