Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S3 / 1 / b / Solution

On the incident side, $\varphi$ is the superposition of incoming and reflected waves, producing a standing-wave pattern whose travelling-wave de Broglie wavelength is $\lambda =2\pi /k$. On the forbidden side, it decays exponentially and remains nonzero near the boundary. A sketch should therefore show oscillations for $x<0$, a continuous function and slope at zero, and the evanescent tail for $x>0$ beside the potential step.

The quantum distinction is finite probability density inside the classically forbidden region, despite complete reflection. There is no transmitted flux at arbitrarily large positive $x$ for this semi-infinite step. A finite-width barrier would allow quantum tunnelling, a different boundary-value problem.