Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 6B / Solution

For the wavefunction of a particle of mass $m$, the probability density and probability current areFor a real potential $V$, the Time-dependent Schrödinger equation and its complex conjugate readMultiplying by ${\Psi }^{\ast }$ and $\Psi$ respectively yieldsThe potential terms cancel because $V$ is real. This is the continuity equation, expressing local conservation of probability.

For the given superposition the common time phase cancels in ${\Psi }^{\ast }{\Psi }_x$, andThe expression in the second parentheses is purely imaginary, so the second term is real and contributes nothing to the probability current. ThusIt is the incoming probability current minus the reflected probability current.