Equal reflection and transmission probabilities

Abstract

We consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as and is finite for all x. We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.

Asymmetric tunneling of Bose–Einstein condensates

“…Generalizing this, Landau gave an intuition as to why tunneling remains symmetric for an arbitrary potential that remains bounded as x → ±∞ (see section 25 of [12]). The tunneling symmetry in a similar case has also been derived from flux conservation in [57] as well as in section 7.3.3 of [58]. In section V of [52] it was shown that symmetry of tunneling is violated for a potential V(x) → const • x as x → ±∞, which is asymptotically an unbounded linear ramp.…”