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

Annalen der Physik

Minimum tunneling time in quantum motion

Christian Bracher, M. Kleber

Published Dec 1, 1998• 3 citations
Publisher

Abstract

Starting out with the dwell time approach by Smith for stationary one-dimensional tunneling problems, we introduce the notion of the minimum tunneling time required for traversal through an arbitrary potential barrier. The newly devised time scale is based on a simple variational principle and presents a strictly local quantity not depending on the asymptotics of the potential distribution. Its properties are illustrated using the customary square barrier as an example. The minimum tunneling time approach cannot support the existence of superluminal velocities in barrier penetration.

DOI
10.1002/(sici)1521-3889(199812)7:7/8<687::aid-andp687>3.0.co;2-1

Related papers

Minimum tunneling time in quantum motion

Christian Bracher, M. Kleber

Abstract Starting out with the dwell time approach by Smith for stationary one‐dimensional tunneling problems, we introduce the notion of the minimum tunneling time required for traversal through an arbitrary potential barrier. The newly devised time scale is based on a simple variational principle and presents a strictly local quantity not depending on the asymptotics of the potential distribution. Its properties are illustrated using the customary square barrier as an example. The minimum tunneling time approach cannot support the existence of superluminal velocities in barrier penetration.

Annalen der Physik•Dec 1, 1998•0 citations

Variational approach to the tunneling-time problem

Christian Bracher, M. Kleber, Mustafa Riza

Tunneling problems are characterized by different quantum time scales of motion. In this paper, we identify a tunneling time scale, which is based on a simple variational principle. The method utilizes the stationary eigenfunctions for a given one-dimensional potential structure, and it provides a truly local definition of the tunneling time, independent of the asymptotic shape of the potential. We express the minimum tunneling time in terms of the more common time scales obtained from the Larmor clock setup. Asymptotic formulas for both the extreme quantum and the semiclassical limit are presented. As an experimental verification of the variational approach we demonstrate that the minimum tunneling time governs the time a particle requires to traverse the barrier in a symmetric double-well structure.

Physical Review A•Sep 1, 1999•19 citations

Quantum Theory of Tunneling

Mohsen Razavy

A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.

WORLD SCIENTIFIC eBooks•Jan 1, 2003•259 citations

Tunneling and group velocity in the square potential barrier

Paolo Moretti

Extending the results obtained in previous papers, an asymptotic analysis is applied to the problem of the penetration of a square potential barrier; it is shown that also in the classically forbidden case a velocity can be attributed to the particle, analogous to the usual group velocity and determining the traversal time.

Physical Review A•Aug 1, 1992•10 citations

Quantum-mechanical tunneling time and its relation to the Tsu-Esaki formula

Marc M. Cahay, T. Dichiaro, P. Thanikasalam +1 more

Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 3, 1992•3 citations

Traversal time for Dirac particles through a potential barrier

Chun-Fang Li, Xi Chen

A traversal time that has no problem of superluminality was advanced for particles to tunnel through potential barriers in the non-relativistic quantum theory in a previous paper by C.-F. Li and Q. Wang, Physica B 296 (2001) 356. This time is generalized in this paper to Dirac's relativistic quantum theory. Both evanescent and propagating cases are considered. It is shown that the traversal time in the evanescent case has much the same properties as in the non-relativistic quantum theory and thus has no problem of superluminality. It also gets rid of the problem of superluminality in the propagating case. Comparisons with the dwell time, the group delay, and the velocity of monochromatic front are also made.

Annalen der Physik•Dec 1, 2002•9 citations
Starting out with the dwell time approach by Smith for stationary one-dimensional tunneling problems, we introduce the notion of the minimum tunneling time required for traversal through an arbitrary potential barrier. The newly devised time scale is based on a simple variational principle and presents a strictly local quantity not depending on the asymptotics of the potential distribution. Its properties are illustrated using the customary square barrier as an example. The minimum tunneling time approach cannot support the existence of superluminal velocities in barrier penetration.
Abstract Starting out with the dwell time approach by Smith for stationary one‐dimensional tunneling problems, we introduce the notion of the minimum tunneling time required for traversal through an arbitrary potential barrier. The newly devised time scale is based on a simple variational principle and presents a strictly local quantity not depending on the asymptotics of the potential distribution. Its properties are illustrated using the customary square barrier as an example. The minimum tunneling time approach cannot support the existence of superluminal velocities in barrier penetration.
Tunneling problems are characterized by different quantum time scales of motion. In this paper, we identify a tunneling time scale, which is based on a simple variational principle. The method utilizes the stationary eigenfunctions for a given one-dimensional potential structure, and it provides a truly local definition of the tunneling time, independent of the asymptotic shape of the potential. We express the minimum tunneling time in terms of the more common time scales obtained from the Larmor clock setup. Asymptotic formulas for both the extreme quantum and the semiclassical limit are presented. As an experimental verification of the variational approach we demonstrate that the minimum tunneling time governs the time a particle requires to traverse the barrier in a symmetric double-well structure.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.
A traversal time that has no problem of superluminality was advanced for particles to tunnel through potential barriers in the non-relativistic quantum theory in a previous paper by C.-F. Li and Q. Wang, Physica B 296 (2001) 356. This time is generalized in this paper to Dirac's relativistic quantum theory. Both evanescent and propagating cases are considered. It is shown that the traversal time in the evanescent case has much the same properties as in the non-relativistic quantum theory and thus has no problem of superluminality. It also gets rid of the problem of superluminality in the propagating case. Comparisons with the dwell time, the group delay, and the velocity of monochromatic front are also made.