Wave and particle limit for multiple barrier tunneling

Abstract

The particle approach to one-dimensional potential scattering is applied to non relativistic tunnelling between two, three and four identical barriers. We demonstrate as expected that the infinite sum of particle contributions yield the plane wave results. In particular, the existence of resonance/transparency for twin tunnelling in the wave limit is immediately obvious. The known resonances for three and four barriers are also derived. The transition from the wave limit to the particle limit is exhibit numerically. I. INTRODUCTION.The particle approach to piece-wise potential scattering is characterized by considering the reflection and transmission amplitudes, successively, at each potential discontinuity. this method, applied also in optics [1], can be used as an alternative to plane wave continuity equations for the derivation of the transmission and reflection amplitudes and find application in approximate solutions to onedimensional potential problems [2,3]. Several interesting papers have analyzed tunneling with this method [4][5][6]. However, in certain situations, this approach is the natural physical choice in the "particle limit" in which the incoming wave packets are small compared to the potentials extension. However, it must be emphasized that the approach itself does not require the use of wave packets. It predicts multiple reflections and even for a single incoming wave it generally results in infinite reflected and transmitted waves. An example of this approach is the diffusion above a single potential barrier, from which one significant consequence, otherwise mysterious, is the step limit for large barrier [7]. The first barrier reflection coefficient reproduces the step result.The alternative and standard approach to potential scattering is with a single wave analysis [8]. We refer to this as the "wave limit". It is characterized by continuity equations often best described by the use of matrices and yields a single reflected and transmitted amplitude. Due to the fact that the algebraic sum of the infinite contributions in the particle approach yields the wave result, the two methods are mathematically equivalent. However, they are not equivalent in practice because a sum of calculated terms, if finite, is an unambiguous process, but the decomposition of an expression as an infinite sum, on the other hand, is highly ambiguous, even if only one decomposition can lead to probability conservation and to the correct exit times of the separate wave packets. In the wave limit, probabilities are calculated by first summing the particle terms and then squaring the modulus. In the particle limit, the probabilities are calculated by first squaring the modulus of each term and then summing.Resonance phenomena, within the Schrödinger equation, are well known for a single barrier when considering plane waves with energy above the potential value, i.e. E > V 0 . It results in unit transmission probabilities. For a given plane wave energy there are unlimited such resonances as the barrier length increases. Mo...