  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.
