WKB propagators in position and momentum space for a linear potential with a "ceiling" boundary

TL;DR

This paper develops WKB propagators for a linear potential with a boundary, analyzing classical paths, caustics, and comparing position and momentum space approaches through numerical tests.

Contribution

It introduces a novel WKB propagator construction for a linear potential with a boundary and compares position and momentum space methods.

Findings

The boundary creates a caustic where amplitude vanishes.

The momentum-space propagator performs better in classically forbidden regimes.

Classical paths and caustics are thoroughly characterized.

Abstract

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 $t$).