Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0411113

Analyses of reflection and transmission at moving potential step

Samura, Toshiharu, Masato Ohmukai

Published Nov 16, 2004• 0 citations
arXivHTMLPDF

Abstract

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 $v$ 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 $V_0$. The transmission occurs when the kinetic energy of incident wave function is larger than the effective potential hight of $(\sqrt{\frac{m}{2}}v + \sqrt{V_0})^2$. The other result is that the reflectivity depends on $x$, which derives from the interference between the incident and the reflected wave functions.

DOI
10.48550/arxiv.quant-ph/0411113

Related papers

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

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.

Canadian Journal of Physics•Oct 2, 2015•1 citation

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon potentials with large changes

Mark R. A. Shegelski, Kevin Malmgren

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.

Canadian Journal of Physics•Jun 11, 2015•3 citations

Quantum Mechanical Potential Step Functions, Barriers, Wells and the Tunneling Effect

Gilbert A. Ibitola, Olanrewaju Ajanaku

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.

World Journal of Applied Physics•Dec 22, 2016•0 citations

Note on the Transmission and Reflection of Wave Packets by Potential Barriers

L. A. MacColl

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 $V(x)$ exceeds the value of the total energy $E$, 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 $V(x)$ is taken as follows: ${V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}$ A wave function is set up which initially represents a wave packet moving toward the point $x=0$ 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 $x=a$ at about the time at which the incident packet reaches the point $x=0$, so that there is no appreciable delay in the transmission of the packet through the barrier.

Physical Review•May 15, 1932•349 citations

Paradoxical Reflection in Quantum Mechanics

Pedro L. Garrido, Sheldon Goldstein, Jani Lukkarinen +1 more

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schrödinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. PACS: 03.65.-w, 03.65.Nk, 01.30.Rr. Key words: Schrödinger equation; potential step; confining potential; reflection and transmission coefficients. 1

•Jan 1, 2008•17 citations

Quantum tunneling and reflection of a molecule with a single bound state

Jeremy J. Kavka, Danielle Kerbrat, Mark R. A. Shegelski

In this article, we present the results of studies on the quantum mechanical tunneling and reflection of a diatomic, homonuclear molecule with a single bound state incident upon a potential barrier. In the first study, we investigate the tunneling of a molecule using a time-dependent formulation. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that a molecule may transition between the bound state and an unbound state numerous times during the process of reflection from or transmission past the barrier. It is also found that, in addition to reflecting and transmitting, the molecule may also temporarily straddle the potential barrier in an unbound state. In the second study, we consider the case of a molecule incident in the bound state upon a step potential with energy less than the step. We show that in the limit where the binding energy ${e}_{0}$ approaches zero and the step potential ${V}_{0}$ goes to infinity, the molecule cannot remain in a bound state if the center of mass gets closer to the step than an arbitrarily large distance ${x}_{0}$ which increases as the magnitude of ${e}_{0}$ decreases, as ${V}_{0}$ increases, or both. We also show that, for ${e}_{0}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ and ${V}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, if the molecule is incident in the bound state, it is reflected in the bound state with probability equal to unity, when the center of mass reaches the reflection distance ${x}_{0}$. We verify that the unbound states exhibit the expected physical behavior. We discuss some surprising results. Connections between our results and investigations done in cold atoms, excitons, Cooper pairs, and Rydberg atoms are discussed.

Physical Review A•Feb 9, 2010•12 citations
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.
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.
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.
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.
This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schrödinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. PACS: 03.65.-w, 03.65.Nk, 01.30.Rr. Key words: Schrödinger equation; potential step; confining potential; reflection and transmission coefficients. 1
In this article, we present the results of studies on the quantum mechanical tunneling and reflection of a diatomic, homonuclear molecule with a single bound state incident upon a potential barrier. In the first study, we investigate the tunneling of a molecule using a time-dependent formulation. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that a molecule may transition between the bound state and an unbound state numerous times during the process of reflection from or transmission past the barrier. It is also found that, in addition to reflecting and transmitting, the molecule may also temporarily straddle the potential barrier in an unbound state. In the second study, we consider the case of a molecule incident in the bound state upon a step potential with energy less than the step. We show that in the limit where the binding energy e0{e}_{0}e0​ approaches zero and the step potential V0{V}_{0}V0​ goes to infinity, the molecule cannot remain in a bound state if the center of mass gets closer to the step than an arbitrarily large distance x0{x}_{0}x0​ which increases as the magnitude of e0{e}_{0}e0​ decreases, as V0{V}_{0}V0​ increases, or both. We also show that, for ${e}_{0}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ and ${V}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, if the molecule is incident in the bound state, it is reflected in the bound state with probability equal to unity, when the center of mass reaches the reflection distance x0{x}_{0}x0​. We verify that the unbound states exhibit the expected physical behavior. We discuss some surprising results. Connections between our results and investigations done in cold atoms, excitons, Cooper pairs, and Rydberg atoms are discussed.