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

American Journal of Physics

Quantal tunneling through a rectangular barrier using ‖ ψ ‖2 and flux

James E. Draper

Published Sep 1, 1980• 3 citations
Publisher

Abstract

The transmission coefficient for quantal tunneling through a rectangular, one-dimensional potential barrier is analyzed. By employing continuity of ‖ ψ ‖2 and flux the results, even with an asymmetrical barrier, are obtained easily. The asymmetrical barrier has the same transmission in either direction. Considerable insight into the form of the most physical quantity, ‖ ψ(x) ‖2, is afforded with little algebra. Graphs of transmission are given for general barriers, and properties of the solutions are discussed.

DOI
10.1119/1.12013

Related papers

Use of ‖Ψ‖2 and flux to simplify analysis of transmission past rectangular barriers or wells

James E. Draper

The transmission coefficient for a completely general one-dimensional rectangular potential barrier or well, or series of steps, is analyzed simply by concentrating on ‖Ψ‖2 rather than Ψ. Graphs of ‖Ψ‖2 for all cases are given, whereas only two textbooks graph anything related to Ψ, and those plots of Re(Ψ) can be misinterpreted.

American Journal of Physics•Jun 1, 1979•7 citations

Quantum-Mechanical Tunneling

J. H. Fermor

An elementary method is given for the exact determination of the quantum mechanical transmission and reflection coefficients for a rectangular one-dimensional potential barrier. The method also finds application in approximate solutions for multistep barriers.

American Journal of Physics•Dec 1, 1966•39 citations

The discussion of the transmission coefficient of one-dimensional double barrier

Jiajun Li

Quantum theory plays a significant part in the modern physics and makes modern technology rapidly developed. Quantum tunnel effect is one of the most important discoveries. The exact expression of potential barrier is tough to derive, even for the double barrier. Many researchers have analysed the applications of quantum tunnel effect in real electronic devices. However, the exact expression of double barrier’s transmission coefficient was discussed rarely. In this article, the exact expression of the transmission coefficient will be given and the effect of parameters of barrier such as the width and the height will be discussed. In addition, the difference between symmetric and asymmetric double barrier will be given. This paper has the discussion of the transmission coefficient in order to not only give a clear understanding of the effect of parameters of double barrier to tyro and researchers, but also give a simpler way to consider the periodic barrier.

Theoretical and Natural Science•Dec 19, 2023•0 citations

Reflection of wave packets from a quantum well with a tunneling transmission resonance

A Gomez Edgar

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.

American Journal of Physics•Feb 1, 1995•8 citations

Coefficients of Abovebarrier and Subbarrier Transmission for a Family of One-Dimensional Structured Barriers

Maria Tashkova, Stoil Donev

We use the transfer-matrix approach to solve the one-dimensional Schrödinger equation for a particular family of structured barriers and to compute in an explicit form the abovebarrier and subbarrier transmission coefficients as functions of barrier parameters. For rectangular barriers the reflectionless conditions are solved and the admissible energy expressions are obtained. For “opaque” barriers the influence of the different barrier factors on the tunneling coefficients is explicitly shown.

physica status solidi (b)•Mar 1, 1997•1 citation

Tunneling-time probability distribution

Randall S. Dumont, T. L. Marchioro II

Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.

Physical Review A•Jan 1, 1993•80 citations
The transmission coefficient for quantal tunneling through a rectangular, one-dimensional potential barrier is analyzed. By employing continuity of ‖ ψ ‖2 and flux the results, even with an asymmetrical barrier, are obtained easily. The asymmetrical barrier has the same transmission in either direction. Considerable insight into the form of the most physical quantity, ‖ ψ(x) ‖2, is afforded with little algebra. Graphs of transmission are given for general barriers, and properties of the solutions are discussed.
Quantum theory plays a significant part in the modern physics and makes modern technology rapidly developed. Quantum tunnel effect is one of the most important discoveries. The exact expression of potential barrier is tough to derive, even for the double barrier. Many researchers have analysed the applications of quantum tunnel effect in real electronic devices. However, the exact expression of double barrier’s transmission coefficient was discussed rarely. In this article, the exact expression of the transmission coefficient will be given and the effect of parameters of barrier such as the width and the height will be discussed. In addition, the difference between symmetric and asymmetric double barrier will be given. This paper has the discussion of the transmission coefficient in order to not only give a clear understanding of the effect of parameters of double barrier to tyro and researchers, but also give a simpler way to consider the periodic barrier.
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.
We use the transfer-matrix approach to solve the one-dimensional Schrödinger equation for a particular family of structured barriers and to compute in an explicit form the abovebarrier and subbarrier transmission coefficients as functions of barrier parameters. For rectangular barriers the reflectionless conditions are solved and the admissible energy expressions are obtained. For “opaque” barriers the influence of the different barrier factors on the tunneling coefficients is explicitly shown.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.