As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point $y$ to point $x$ in time $t$ and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usual sense when the reflecting boundary is approximated by steeply rising smooth potentials. Then, to improve the WKB approximation we construct the propagator for initial data in momentum space; this requires classifying the interesting variety of classical paths with initial momentum $p$ arriving at $x$ after time $t$. The two approximate propagators are compared by applying them to Gaussian initial packets by numerical integration; the results show physically expected behavior, with advantages to the momentum-based propagator in the classically forbidden regime (large $t$).
Download