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:0910.4564 · American Journal of Physics

Reflection above the barrier as tunneling in momentum space

R. L. Jaffe

Published May 12, 2010• 14 citations
arXivHTMLPDF

Abstract

Quantum mechanics predicts an exponentially small probability that a particle with energy greater than the height of a potential barrier will reflect from the barrier in violation of classical expectations. This process can be regarded as tunneling in momentum space, leading to a simple derivation of the reflection probability.
DOI
10.1119/1.3298428

Related papers

Barrier Tunneling and Reflection in the Time and Energy Domains: The Battle of the Exponentials

Neepa T. Maitra, Eric J. Heller

The issue of quantum barrier crossing and reflection in the time domain is addressed. We find that (1) classically forbidden barrier tunneling and above-barrier reflection are well-defined and important processes in the time domain, (2) classically forbidden processes can overshadow allowed ones when both are present, and (3) classically allowed trajectories in the time domain are not, in general, sufficient to explain tunneling amplitudes in the energy domain. We also make clear the essential distinction of barriers which flatten out at large distance and those which do not.

Physical Review Letters•Apr 21, 1997•88 citations

Quantum Mechanical Example of Anti-Tunneling

T.B. Mitchell

The probability of a particle being reflected by a semi-infinite energy barrier was solved using Schrödinger's equations that describe the quantum mechanical behavior of the particle. The continuity of the functions at the origin allowed the wave functions and their respective derivatives to be equated. Solution of the resulting system yielded values A1 and B1 which were used to calculate the reflection coefficient, (B1/A1)2. The conclusions obtained from these results stated that the particle would be reflected by the barrier when E

Undergraduate Journal of Mathematical Modeling One + Two•May 1, 2013•0 citations

Quantum reflection shields ultracold atoms

Harald Friedrich

When a ball rolls down a hill, the last thing you expect is for it to suddenly stop and then roll straight back up the hill. At the atomic scale, however, the wave nature of particles becomes important and can lead to a counterintuitive phenomenon called quantum reflection. This is a purely quantum-mechanical process in which a particle moving in a given direction can reverse its direction of motion without reaching a classical turning point. It is thus the counterpart of quantum tunnelling, in which a particle can penetrate a potential barrier in a way that is forbidden by classical mechanics.

Physics World•Aug 1, 2004•2 citations

A Classical Explanation of Quantum Tunneling: An Energy Transfer Model Based on Electron Collisions

Fan Xie

This paper proposes a classical alternative explanation for the tunneling effect in quantum mechanics. Standard quantum theory describes tunneling as a particle "penetrating" a potential barrier whose energy exceeds the particle's own kinetic energy with a certain probability. However, this explanation focuses only on the mathematical initial and final states, completely ignoring any dynamical process the electron may undergo before the barrier, and lacks a physical picture. This paper argues that tunneling is not an intrinsic quantum property of a single electron, but rather a collective statistical fluctuation event involving multiple electrons: electrons near the barrier, through a series of random collisions in the correct direction, gradually transfer and concentrate kinetic energy onto the electron closest to the barrier, enabling it to instantaneously acquire kinetic energy exceeding the barrier threshold and thus classically fly over the barrier. During the crossing, the electron must do work against the barrier, converting kinetic energy into potential energy; therefore, after crossing, its kinetic energy is reduced and it cannot cross the same barrier again. Tunneling probability is reinterpreted as the success probability of such a "directed energy relay" event. This model is fully self-consistent within the framework of classical electromagnetism and statistical mechanics, requiring no wave functions or probability waves, with a clear physical picture and explicit causal chain. This paper further points out that under strictly single-electron conditions, this model predicts that tunneling will completely disappear, a prediction that stands in fundamental opposition to the standard quantum model and can serve as a decisive experiment to test the two theories.

•Aug 13, 2026•0 citations

Induced Optical Tunnelling

C. Hirlimann, Benjamin Thomas

François Bardou and Dominique Boosé recently published1 their discovery of a new induced tunnel effect, they named quantum evaporation. In this work they showed that, for a one-dimensional electron reflecting on an energy barrier, the stationary character of its eigen-functions could be broken under special conditions. If a small momentum is transferred to the electron at the time it reflects on the barrier, its wave-function is projected on the complete set of its stationary eigen-functions. A small momentum means that the energy transfer that accompanies the momentum transfer is negligible compared to the barrier height or at least that it does not allow for a jump of the particle above the barrier. The tunnelling probability of the electron is strongly enhanced by the momentum perturbation because the weight of the eigen-states with energies above the barrier becomes preponderant in the calculation of the tunnelling probability. This probability enhancement is independent on the sign of the momentum transfer: the electron can be pushed toward or pull out of the barrier; it only depends on the momentum amplitude in a quadratic way. The authors have showed that the momentum transfer is equivalent to a kinetic translation of the barrier.

International Conference on Ultrafast Phenomena•Jan 1, 2002•0 citations

A Study Of Quantum Tunneling Through Potential Barriers

Akshay Bhatia

This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.

•Jul 24, 2026•0 citations
The issue of quantum barrier crossing and reflection in the time domain is addressed. We find that (1) classically forbidden barrier tunneling and above-barrier reflection are well-defined and important processes in the time domain, (2) classically forbidden processes can overshadow allowed ones when both are present, and (3) classically allowed trajectories in the time domain are not, in general, sufficient to explain tunneling amplitudes in the energy domain. We also make clear the essential distinction of barriers which flatten out at large distance and those which do not.
The probability of a particle being reflected by a semi-infinite energy barrier was solved using Schrödinger's equations that describe the quantum mechanical behavior of the particle. The continuity of the functions at the origin allowed the wave functions and their respective derivatives to be equated. Solution of the resulting system yielded values A1 and B1 which were used to calculate the reflection coefficient, (B1/A1)2. The conclusions obtained from these results stated that the particle would be reflected by the barrier when E
When a ball rolls down a hill, the last thing you expect is for it to suddenly stop and then roll straight back up the hill. At the atomic scale, however, the wave nature of particles becomes important and can lead to a counterintuitive phenomenon called quantum reflection. This is a purely quantum-mechanical process in which a particle moving in a given direction can reverse its direction of motion without reaching a classical turning point. It is thus the counterpart of quantum tunnelling, in which a particle can penetrate a potential barrier in a way that is forbidden by classical mechanics.
This paper proposes a classical alternative explanation for the tunneling effect in quantum mechanics. Standard quantum theory describes tunneling as a particle "penetrating" a potential barrier whose energy exceeds the particle's own kinetic energy with a certain probability. However, this explanation focuses only on the mathematical initial and final states, completely ignoring any dynamical process the electron may undergo before the barrier, and lacks a physical picture. This paper argues that tunneling is not an intrinsic quantum property of a single electron, but rather a collective statistical fluctuation event involving multiple electrons: electrons near the barrier, through a series of random collisions in the correct direction, gradually transfer and concentrate kinetic energy onto the electron closest to the barrier, enabling it to instantaneously acquire kinetic energy exceeding the barrier threshold and thus classically fly over the barrier. During the crossing, the electron must do work against the barrier, converting kinetic energy into potential energy; therefore, after crossing, its kinetic energy is reduced and it cannot cross the same barrier again. Tunneling probability is reinterpreted as the success probability of such a "directed energy relay" event. This model is fully self-consistent within the framework of classical electromagnetism and statistical mechanics, requiring no wave functions or probability waves, with a clear physical picture and explicit causal chain. This paper further points out that under strictly single-electron conditions, this model predicts that tunneling will completely disappear, a prediction that stands in fundamental opposition to the standard quantum model and can serve as a decisive experiment to test the two theories.
François Bardou and Dominique Boosé recently published1 their discovery of a new induced tunnel effect, they named quantum evaporation. In this work they showed that, for a one-dimensional electron reflecting on an energy barrier, the stationary character of its eigen-functions could be broken under special conditions. If a small momentum is transferred to the electron at the time it reflects on the barrier, its wave-function is projected on the complete set of its stationary eigen-functions. A small momentum means that the energy transfer that accompanies the momentum transfer is negligible compared to the barrier height or at least that it does not allow for a jump of the particle above the barrier. The tunnelling probability of the electron is strongly enhanced by the momentum perturbation because the weight of the eigen-states with energies above the barrier becomes preponderant in the calculation of the tunnelling probability. This probability enhancement is independent on the sign of the momentum transfer: the electron can be pushed toward or pull out of the barrier; it only depends on the momentum amplitude in a quadratic way. The authors have showed that the momentum transfer is equivalent to a kinetic translation of the barrier.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.