Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 1 / 3 / Solution

As $x\to -\infty$, the scattered kernel is $e^{-ikx}{e}^{ik{x}^{\prime }}$; as $x\to +\infty$, it is $e^{ikx}{e}^{-ik{x}^{\prime }}$. Therefore ${\psi }_s\sim e^{ikx}+r{e}^{-ikx}$ on the left and $t{e}^{ikx}$ on the right, withEqual asymptotic wavenumbers give $R=|r{|}^2$, $T=|t{|}^2$. Squaring the first amplitude proves the requested expression. These are current ratios, which would require a velocity factor if the two asymptotic potentials differed.