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Canadian Journal of Physics

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon a step potential and a square potential well

Mark R. A. Shegelski, Kevin Malmgren, Logan Salayka-Ladouceur

Published Oct 2, 2015• 1 citation
Publisher

Abstract

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.

DOI
10.1139/cjp-2015-0569

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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.
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 and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a constant velocity vvv in the same direction as an incident wave function by means of solving the time-dependent Schrödinger equation. As for an infinite potential step, it is known that group velocity is the same as the moving velocity of the potential step. We found two interesting results when the potential step has a finite height of V0V_0V0​. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of (m2v+V0)2(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2(2m​​v+V0​​)2. The other result is that the reflectivity depends on xxx, which derives from the interference between the incident and the reflected wave functions.
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.
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 x → ± ∞ 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.
In this paper, use is made of the tools of analytical mechanics and the concept of operators to obtain the time-independent and time-dependent Schrodinger wave equations for quantum mechanical systems. Derivations are embarked upon of expressions for reflection and transmission coefficients for a particle of mass m as well as of energy E moving under different potential set-ups across step functions, barriers and well functions. The tunneling effect is then discussed. The transmission probability equation obtained in this research has been observed to be more accurate than the transmission probability expression deduced by some researchers in 2014 for a tunneling barrier. This research work finds applications in nuclear magnetic resonance imaging systems, synchrotrons, gyrators, accelerators, and in electrodynamics.
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