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

Journal of Applied Physics

Distribution of times for barrier traversal caused by energy fluctuations

Mark J. Hagmann

Published Dec 15, 1993• 5 citations
Publisher

Abstract

Quantum tunneling is attributed to energy fluctuations permitted by the uncertainty principle. The distribution of barrier traversal times is shown to be bimodal. The expression for the expectation value is similar to the semiclassical result, and the width of the distribution varies inversely with the size of the barrier.
DOI
10.1063/1.354995

Related papers

Distribution of barrier traversal times in numerical simulations

Mark J. Hagmann

Barrier penetration is attributed to energy fluctuations expected from the uncertainty principle. Numerical simulations are made by calculating the traversal time and action for a large number of possible velocity profiles. Distributions of traversal time are determined by assuming that the probability of each velocity profile decreases exponentially with the action of the fluctuation it requires. Distributions of traversal times are reported for rectangular barriers having different sizes. For large barriers the distributions are leptokurtic and centered at the semiclassical traversal time T0 = d√m/[2(V0−E)], where d and V0 are the length and height of the barrier and m and E are the mass and energy of the particle. The kurtosis decreases and the mode shifts to shorter durations with decreasing barrier size.

Journal of Applied Physics•Aug 1, 1993•3 citations

Path decomposition and the traversal-time distribution in quantum tunneling

H. A. Fertig

We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2${\mathit{md}}^{2}$/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.

Physical review. B, Condensed matter•Jan 15, 1993•33 citations

Traversal-Time Distribution and the Uncertainty Principle in Quantum Tunneling

H. A. Fertig

We derive the exact distribution of times spent in the classically forbidden region by a particle as it tunnels through a square barrier potential. We find that the integrated distribution has a simple interpretation. If one restricts the time spent in the barrier to be \ensuremath{\leqslant}\ensuremath{\tau}, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that\ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\simeq}1. This uncertainty leads to unusual interference effects in the distribution. These interference effects explain why the average speed of a particle traversing a very opaque barrier may exceed the speed of light. \textcopyright{} 1990 The American Physical Society

Physical Review Letters•Nov 5, 1990•123 citations

Barrier-traversal-time operator and the time-energy uncertainty relation

Denny Lane B. Sombillo, Eric A. Galapon

A general barrier-traversal-time operator is constructed using the time-of-arrival formalism. We study the operator's dynamics and determine the role played by the time-energy commutation relation. It turns out that similar dynamics is observed whether the traversal-time operator is canonically conjugate to the system Hamiltonian or not. We also use the barrier-traversal-time operator to calculate the traversal-time distributions for different cases (free case, and above-the-barrier and under-the-barrier traversals). The peak of the traversal-time distributions coincides with the classical expected traversal times for the free case and the above-the-barrier case. We then present our interpretation of the time-energy uncertainty relation that is consistent with the different traversal-time distributions.

Physical Review A•Jun 27, 2018•11 citations

A Simple Treatment of Potential Barrier Penetration

Bernard L. Cohen

It is pointed out that the penetration of a particle through a potential barrier may be understood as a climb over the barrier during one of the energy fluctuations expected from the uncertainty principle, rather than as a “tunneling” under the barrier. A method of obtaining the quantitative result with this model is presented.

American Journal of Physics•Feb 1, 1965•21 citations

Early times in tunneling

Gastón García‐Calderón, Jorge Villavicencio

Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak $τ_p$ and a width $Δτ$. Numerical results for other tunneling times, as the phase-delay time, fall within $Δτ$. The Büttiker traversal time is the closest to $τ_p$. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.

arXiv:quant-ph/0008014•Aug 3, 2000•0 citations
Barrier penetration is attributed to energy fluctuations expected from the uncertainty principle. Numerical simulations are made by calculating the traversal time and action for a large number of possible velocity profiles. Distributions of traversal time are determined by assuming that the probability of each velocity profile decreases exponentially with the action of the fluctuation it requires. Distributions of traversal times are reported for rectangular barriers having different sizes. For large barriers the distributions are leptokurtic and centered at the semiclassical traversal time T0 = d√m/[2(V0−E)], where d and V0 are the length and height of the barrier and m and E are the mass and energy of the particle. The kurtosis decreases and the mode shifts to shorter durations with decreasing barrier size.
We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2md2{\mathit{md}}^{2}md2/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.
We derive the exact distribution of times spent in the classically forbidden region by a particle as it tunnels through a square barrier potential. We find that the integrated distribution has a simple interpretation. If one restricts the time spent in the barrier to be \ensuremath{\leqslant}\ensuremath{\tau}, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that\ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\simeq}1. This uncertainty leads to unusual interference effects in the distribution. These interference effects explain why the average speed of a particle traversing a very opaque barrier may exceed the speed of light. \textcopyright{} 1990 The American Physical Society
A general barrier-traversal-time operator is constructed using the time-of-arrival formalism. We study the operator's dynamics and determine the role played by the time-energy commutation relation. It turns out that similar dynamics is observed whether the traversal-time operator is canonically conjugate to the system Hamiltonian or not. We also use the barrier-traversal-time operator to calculate the traversal-time distributions for different cases (free case, and above-the-barrier and under-the-barrier traversals). The peak of the traversal-time distributions coincides with the classical expected traversal times for the free case and the above-the-barrier case. We then present our interpretation of the time-energy uncertainty relation that is consistent with the different traversal-time distributions.
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.