= Solution
On the incident side, $\phi$ 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 \b[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.
\Image[/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-potential-step.png]
{title=A reflected wave joins a decaying tail inside a semi-infinite potential step}
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