Reflection above the barrier as tunneling in momentum space

a potential V (x), when the height of the potential exceeds the particle’s energy for a < x < b. Elementary derivations of this result can be found in most textbooks[1]. Quantum mechanics also predicts that a particle with energy E, greater than a barrier V (x) for all x, will reflect from the barrier. This can also be estimated using semiclassical methods. A derivation can be found in Landau and Lifschitz’s well-known presentation of the semiclassical method[2], but the derivation is more complicated and less intuitive than the derivation of the barrier penetration factor. In this paper we show that reflection above the barrier can be understood as barrier penetration in momentum space, making the physical origin of the phenomenon more obvious and reducing the derivation to that of ordinary barrier penetration. This appears not to have been noticed in the past. An added benefit of this view of reflection above the barrier is that it leads to a simple derivation of the transition probability in the adiabatic approximation — the Landau-Zener formula — again based only on the physics of barrier penetration. The paper is organized as follows: In Section II the problem of reflection above the barrier is shown to be barrier penetration in momentum space. In Section III the result is shown to be identical to the result derived in Ref. [2]. In Section IV a few simple examples are presented. In Section V the transition probability in the adiabatic approximation is recast as reflection above the barrier in time (as opposed to space) and the Landau-Zener result is derived.