  Let $H_h = h^2 L +V$ where $L$ is a self-adjoint Laplace type operator acting
on sections of a vector bundle over a compact Riemannian manifold and $V$ is a
symmetric endomorphism field. We derive an asymptotic expansion for the heat
kernel of $H_h$ as $h \to 0$. As a consequence we get an asymptotic expansion
for the quantum partition function and we see that it is asymptotic to the
classical partition function. Moreover, we show how to bound the quantum
partition function for positive $h$ by the classical partition function.
