  This article concerns a phenomenon of elementary quantum mechanics that is
quite counter-intuitive, very non-classical, and apparently not widely known: a
quantum particle can get reflected at a downward potential step. In contrast,
classical particles get reflected only at upward steps. The conditions for this
effect are that the wave length is much greater than the width of the potential
step and the kinetic energy of the particle is much smaller than the depth of
the potential step. This phenomenon is suggested by non-normalizable solutions
to the time-independent Schroedinger equation, and we present evidence,
numerical and mathematical, that it is also indeed predicted by the
time-dependent Schroedinger equation. Furthermore, this paradoxical reflection
effect suggests, and we confirm mathematically, that a quantum particle can be
trapped for a long time (though not forever) in a region surrounded by downward
potential steps, that is, on a plateau.
