We consider the probabilities of transmission p T and reflection p R of a quantum particle incident
upon potential drops with significant qualitative differences. We look at a smooth potential drop
for which the potential and the derivative of the potential are continuous. We also consider a
potential drop which is continuous but has discontinuities in its derivative. The two cases give
markedly different results for the limiting values of p T and p R with increasing values of the
total potential drop V 0 . We explore the difference in the results using potentials that can be
adjusted to go from continuous to discontinuous first derivatives. We also investigate potentials
with more severe discontinuities in the derivative of the potential.