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Canadian Journal of Physics

Transmission and reflection of a wave packet with a small width in one-dimensional quantum mechanics

F. Masafumi Toyama, Yukihisa Nogami

Published Oct 1, 2003• 3 citations
Publisher

Abstract

The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk

DOI
10.1139/p03-102

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The transmission and the reflection of a wave incident on a potential in one-dimensional quantum mechanics are usually discussed by assuming that a plane wave eikx or a broad wave packet is incident. We are interested in what happens if the width of the incident wave packet is very small. In this case, the incident wave packet can spread out in such a way that part of the wave packet turns around before reaching the potential area and proceeds backward. This can clearly be visualized by means of the Bohm trajectories associated with the wave packet. We discuss a complication that arises in this case regarding the notion of the reflection probability.PACS Nos.: 03.65–w, 03.65.Nk
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.
We discuss "the plane wave approximation" to quantum mechanical scattering using simple one-dimensional examples. The central points of the paper are that (a) plane waves should be thought of as infinitely wide wave packets, and (b) the calculations of reflection and transmission probabilities RRR and TTT in standard textbook presentations involve an approximation which is almost never discussed. We argue that it should be discussed explicitly, and that doing so provides a simple and intuitively revealing alternative way to derive and understand certain formulas. Using an under-appreciated exact expression for wave packet scattering probabilities, we calculate, for two standard examples, expressions for RRR and TTT for an incident Gaussian wave packet. Comparing these results to the corresponding ones calculated using the plane wave approximation helps illuminate the domain of applicability of that approximation.
We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
In many quantum mechanics courses, plane waves and wave packets are introduced at the very beginning, and used to solve 'simple' one-dimensional problems with square potentials. The purpose of this article is to clarify and criticise the meaning of commonly used statements such as: 'an e ikx term represents a particle moving in the positive x direction' and/or 'e iks represents the incident beam of particles'. The necessary notions and results are first briefly reviewed. A simple problem (scattering by a one-dimensional square potential step) is then studied in detail in order to exhibit and discuss the difficulties in a concrete situation. The more general relevance of this discussion is finally indicated.
We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.