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American Journal of Physics

One-dimensional scattering by a locally periodic potential

Hai-Woong Lee, Adam Zysnarski, Phillip L. Kerr

Published Aug 1, 1989• 27 citations
Publisher

Abstract

One-dimensional scattering of a particle or a wave packet by a finite number of periodic potential barriers is studied. The fundamental physical mechanism that determines the transmission coefficient is shown to be quantum interference between waves with different numbers of inner reflections. The tendency of the transmission coefficient to saturate with respect to an increase in the number of barriers is due to the free-particle nature of a particle subject to a completely periodic potential.

DOI
10.1119/1.16134

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Physical Review•May 15, 1932•349 citations
One-dimensional scattering of a particle or a wave packet by a finite number of periodic potential barriers is studied. The fundamental physical mechanism that determines the transmission coefficient is shown to be quantum interference between waves with different numbers of inner reflections. The tendency of the transmission coefficient to saturate with respect to an increase in the number of barriers is due to the free-particle nature of a particle subject to a completely periodic potential.
We discuss a solution of the time-dependent Schrödinger equation that incorporates absorbing boundary conditions and a method for extracting the reflection and transmission probabilities for wave packets interacting with time-dependent potential barriers. We apply the method to a rectangular barrier that moves with constant velocity, an oscillating rectangular barrier, a locally periodic barrier with an amplitude modulated by a traveling wave, and a locally periodic potential with an amplitude modulated by a standing wave. Visualizations of the reflection phenomena are presented with an emphasis on understanding these systems from their dynamics. Applications to non-stationary neutron optics experiments are discussed briefly.
One-dimensional scattering from a series of delta-function barriers is considered, in the context of nonrelativistic quantum mechanics. A surprisingly simple closed-form expression for the transmission probability is obtained, and compared with the earlier result of Kiang [Am. J. Phys. 42, 785–787 (1974)]. The band structure characteristic of periodic potentials emerges even when the number of barriers is quite small. However, the ‘‘saturation’’ phenomenon identified by Lee, Zysnarski, and Kerr [Am. J. Phys. 57, 729–734 (1989)] is found to depend on the energy regime in question.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.