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

Physica Scripta

On the problem of the quantum mechanical tunneling

Paolo Moretti

Published Jan 1, 1992• 20 citations
Publisher

Abstract

In this paper we try to give an answer to the question: how a particle tunneling through a classically forbidden region behaves? Working out the quantum mechanical analysis in a particular case first proposed by Stevens, we find that the group velocity plays an important role, so confirming the results previously obtained by some authors.

DOI
10.1088/0031-8949/45/1/002

Related papers

Tunneling times

Partha Ghose

Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.

Cambridge University Press eBooks•Apr 29, 1999•0 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 Mechanics as a Classical Theory XIII: The Tunnel Effect

L. S. F. Olavo

In this continuation paper we will address the problem of tunneling. We will show how to settle this phenomenon within our classical interpretation. It will be shown that, rigorously speaking, there is no tunnel effect at all.

arXiv:quant-ph/9609003•Sep 4, 1996•0 citations

Velocity of a Quantum Particle in a Classically Forbidden Region

Christian Beck, Sheldon Goldstein, Dustin Lazarovici +2 more

Recently, Sharoglazova et al. [Nature 643, 67 (2025)] proposed a procedure for determining the speed of a quantum particle in the classically forbidden region of a potential step, and implemented it in a beautiful experiment. The inferred speeds disagree significantly with the Bohmian velocities, which the authors presented as an experimental challenge to Bohmian mechanics. This is puzzling because the speeds are inferred from particle populations in coupled waveguides for which Bohmian mechanics and standard quantum mechanics make identical predictions. We resolve the puzzle by a detailed theoretical analysis of the experimental setup. We show that the speed inference rests on an assumption that fails in the relevant (evanescent) regime according to both Bohmian mechanics and standard quantum mechanics -- namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom. We also consider a second speed estimate suggested by Sharoglazova et al., which is based on the Büttiker dwell time formula for particles in the forbidden region. We show that the authors applied the formula incorrectly, and that a correct application yields exact agreement with the predictions of Bohmian mechanics. Our analysis includes explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in the two-dimensional waveguide model of the experiment.

quant-pharXiv:2607.00100•Jul 2, 2026•0 citations

Barrier Traversal Time in the Quantum Potential Picture

Timothy P. Spiller, Terrence D. Clark, Robert J. Prance +1 more

We discuss the time taken for a particle to pass through a classically forbidden barrier in the quantum potential approach to quantum mechanics.

Europhysics Letters (EPL)•May 1, 1990•41 citations

Some Analytical Results on Classical Tunneling of Sommerfeld Particle

А. А. Власов

A simple example of especially constructed potential barrier enables to show analytically (not numerically) the existence of tunneling effect for a Sommerfeld particle.

•Oct 11, 2001•0 citations
In this paper we try to give an answer to the question: how a particle tunneling through a classically forbidden region behaves? Working out the quantum mechanical analysis in a particular case first proposed by Stevens, we find that the group velocity plays an important role, so confirming the results previously obtained by some authors.
Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.
Recently, Sharoglazova et al. [Nature 643, 67 (2025)] proposed a procedure for determining the speed of a quantum particle in the classically forbidden region of a potential step, and implemented it in a beautiful experiment. The inferred speeds disagree significantly with the Bohmian velocities, which the authors presented as an experimental challenge to Bohmian mechanics. This is puzzling because the speeds are inferred from particle populations in coupled waveguides for which Bohmian mechanics and standard quantum mechanics make identical predictions. We resolve the puzzle by a detailed theoretical analysis of the experimental setup. We show that the speed inference rests on an assumption that fails in the relevant (evanescent) regime according to both Bohmian mechanics and standard quantum mechanics -- namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom. We also consider a second speed estimate suggested by Sharoglazova et al., which is based on the Büttiker dwell time formula for particles in the forbidden region. We show that the authors applied the formula incorrectly, and that a correct application yields exact agreement with the predictions of Bohmian mechanics. Our analysis includes explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in the two-dimensional waveguide model of the experiment.