We have often found among many of our students and colleagues the common idea that the mathematical
expression for a physical quantity that is essentially of quantum nature must contain a dependence
on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference
to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one
of the simplest examples in quantum mechanics, contradicts this common thought: even when the
particle carries enough kinetic energy to go across the step, the resulting expression for the
reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to
ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in
which the potential rises sharply at a single position, thus losing any reference to a length
dimension in the problem. To address the correct classical limit of the phenomenon we first
regularise the discont…