Transmission and reflection of a quantum particle incident upon potential drops

Abstract

We consider the probabilities of transmission pT and reflection pR 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 pT and pR with increasing values of the total potential drop V0. 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.

Equal reflection and transmission probabilities

“…as  ¥ V 0 , which is not zero for nonzero ka. (The difference in these limiting values turns out to be due to a continuous slope for the exponential potential but a discontinuity in slope for the step potential, as reported in [4].) Therefore, we still have that one potential has  p 0 T while the other has p T going to a nonzero value as  ¥ V 0…”

“…x a exp 0 Solving for the probabilities requires the use of hypergeometric functions (as shown in [4]).…”

“…Calculation of these probabilities is significantly more difficult; see, for example, the calculations presented in [4].…”