  We consider the quantum mechanical propagator for a particle moving in a
$d$-dimensional Lorentz gas, with fixed, hard sphere scatterers. To evaluate
this propagator in the semi-classical region, and for times less than the
Ehrenfest time, we express its effect on an initial Gaussian wave packet in
terms of quantities analogous to those used to describe the exponential
separation of trajectories in the classical version of this system. This result
relates the spread of the wave packet to the rate of separation of classical
trajectories, characterized by positive Lyapunov exponents. We consider
applications of these results, first to illustrate the behavior of the
wave-packet auto-correlation functions for wave packets on periodic orbits. The
auto-correlation function can be related to the fidelity, or Loschmidt echo,
for the special case that the perturbation is a small change in the mass of the
particle. An exact expression for the fidelity, appropriate for this
perturbation, leads to an analytical result valid over very long time
intervals, inversely proportional to the size of the mass perturbation. For
such perturbations, we then calculate the long-time echo for semi-classical
wave packets on periodic orbits. This paper also corrects an earlier
calculation for a quantum echo, included in a previous version of this paper.
We explain the reasons for this correction.
