Quantum reflection probability ($R=|r{|}^2$)

For one-dimensional scattering from a real potential, write the left exterior wavefunction as $e^{ikx}+r{e}^{-ikx}$ with $k>0$. The reflected-to-incident probability-current ratio is $R=|r{|}^2$, since both components have the same speed and their currents have opposite signs. This probability differs from the complex reflection amplitude $r$. With transmission into a right exterior wavenumber $q$, the transmission probability is $T=(q/k)|t{|}^2$ and conservation of current gives $R+T=1$ when there is no absorption.

A quantum reflection amplitude is the complex ratio of the outgoing reflected wave to the incident wave in a chosen scattering convention. In one reflected channel with the same exterior speed, the reflection probability is $|r{|}^2$. Its phase affects interference even when its modulus is one; shifting the reference plane changes that phase.

A particle incident below a semi-infinite step has a unit-modulus quantum reflection amplitude but a nontrivial phase. Continuity of the oscillating and evanescent solutions yields the displayed coefficient, for the incident phase origin at the interface. Its decaying exterior wave has zero net probability current. The phase depends on the chosen scattering convention; the unit reflection probability does not.