Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 8B (Quantum Mechanics)

For a particle of mass $m$, the probability density and one-dimensional probability current areThe Schrodinger equation and its complex conjugate, for real $V$, giveThe potential terms cancel precisely because $V$ is real. Thus ${\rho }_t+j_x=0$, the continuity equation for probability. Integrating over an interval says that its probability changes by the incoming current minus the outgoing current.

The global time-dependent phase cancels from the probability current. Put $\chi =e^{ikx}+R{e}^{-ikx}$. Direct multiplication givesThe second term is real, because the expression in parentheses is purely imaginary. HenceThe first plane wave travels to the right with incident current $\hslash k/m$; the second travels to the left with reflected current $-(\hslash k/m)|R{|}^2$. Their wave interference changes the probability density but cancels from the net current. The current vanishes for $|R|=1$, is positive for $|R|<1$, and is negative for $|R|>1$. These plane waves are scattering states, not individually normalizable bound states.