Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Section B / 8 / c

For $E\gg V_0$, the two wavenumbers nearly coincide:Consequently $B/A\simeq V_0/(4E)\to 0$ and $C/A\to 1$. To leading order the complete wavefunction isThe small fractional change in kinetic energy scarcely changes the propagation wavenumber, so reflection vanishes and the particle behaves approximately as a free right-moving plane wave. A more accurate phase at large positive $x$ retains $k_{2}x$ on the transmitted side; even a small wavenumber difference accumulates over a sufficiently long distance.