Classical Path Approximation for the Boltzmann Density Matrix

Using the classical-limit approximation for the quantum mechanical time evolution operator and the formal relation between time and reciprocal temperature (t=−iℏ β, β=1/kT), a ``better than classical'' approximation is obtained for the Boltzmann density matrix. The result involves classical trajectories in a potential which is the negative of the actual potential; it is seen that this effectively allows for some degree of tunneling. This new approximation gives exact quantum results in any region that the potential is quadratic and quite reasonable results for any potential, even in the limit of zero temperature.