Paradoxical Reflection in Quantum Mechanics

TL;DR

This paper explores the counter-intuitive phenomenon where quantum particles reflect off downward potential steps, unlike classical particles, and demonstrates how this effect can trap particles in a potential plateau.

Contribution

It provides a detailed analysis and evidence that quantum particles can reflect from downward steps, revealing a paradoxical effect not widely recognized in quantum mechanics.

Findings

Quantum particles can reflect at downward potential steps.

Particles can be trapped in regions surrounded by downward steps.

The effect is supported by numerical and mathematical evidence.

Abstract

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.