Quantum step-reflection phase ($r=(k-i\kappa )/(k+i\kappa ),\delta =-2\arctan (\kappa /k)$)

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.