Path-Integral Treatment of Quantum Bouncers

Abstract

The one-sided bouncer and the symmetric bouncer involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the one-sided bouncer and symmetric bouncer, which explains why each bounce of the one-sided bouncer increases the Morse index by 2 and results in a phase change of $\pi$. We interpret the semiclassical Feynman path integral to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.