We investigate the time-dependent Schrödinger equation for the case of a plane matter wave incident on an oscillating potential step. Approximate solutions are given for different regimes of the problem. The resulting energy spectra are quantized. This may either be interpreted as a phase modulation of the wave or as a coherent multiphonon exchange. A comparison to the results that apply for a beam of classical particles is made. In particular, the transition from the quantum-mechanical to the classical case is examined. We have performed an experiment with very cold neutrons that clearly revealed the discussed effects. Energy splittings of 2.8 to 9.1 neV of the reflected waves were analyzed with an energy resolution of ∼1 neV, thus demonstrating the feasibility of such high-resolution experiments with neutron reflectometry. © 1996 The American Physical Society.

• Received 6 June 1995

DOI:https://doi.org/10.1103/PhysRevA.53.319