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

Physical Review

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

L. A. MacColl

Published May 15, 1932• 349 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.40.621

Related papers

Tunneling of a Wave Packet

Thomas E. Hartman

Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.

Journal of Applied Physics•Dec 1, 1962•821 citations

Wave functions and tunneling times for one-dimensional transmission and reflection

Nikolay Leonidovich Chuprikov

It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.

arXiv:quant-ph/0311090•Nov 13, 2003•1 citation

A time-dependent study of resonant tunneling through a double barrier

Andrew P. Stamp, Gregory C. McIntosh

The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.

American Journal of Physics•Mar 1, 1996•19 citations

Reflections on a Wave Packet Approach to Quantum Mechanical Barrier Penetration

Mark H. Bramhall, Barry M. Casper

We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.

American Journal of Physics•Sep 1, 1970•29 citations

Analyses of reflection and transmission at moving potential step

Samura, Toshiharu, Masato Ohmukai

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.

arXiv:quant-ph/0411113•Nov 16, 2004•0 citations

Multiple Reflection in Potential-Barrier Scattering

John E. Beam

The standard quantum mechanical problem of scattering of an incident plane wave by a rectangular potential barrier in one dimension is examined from a point of view which emphasizes the multiple-scattering contributions to the reflected and transmitted, waves. This approach is used to gain insight into the phenomenon of resonant scattering of a wave packet by a potential barrier.

American Journal of Physics•Dec 1, 1970•12 citations
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)exp(\ensuremath−2\ensuremathπiEth)f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})f(x,E)exp(\ensuremath−h2\ensuremathπiEt​). 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.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.
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.