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

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Resonance tunneling of a classical particle

Vadim Berdichevsky, Moshe Gitterman

Published Jan 1, 1996• 5 citations
Publisher

Abstract

Exact solutions are obtained for the diffusion in single- and square-well potentials, by using the Laplace transform method. We compare transmission of a classical particle through one barrier of given length and through two barriers with a well between them of the same overall length. It turns out that the existence of a well facilitates the transmission of a classical particle for intermediate times, which bears some resemblance to resonances tunneling through quantum barriers. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.1250

Related papers

Resonant tunnelling of a composite particle through a single potential barrier

Noriko Saito, Yōsuke Kayanuma

Some aspects of quantum tunnelling of a composite particle are clarified by a simple model. We calculate the probability that a couple of point particles bound to each other by a square well potential with a hard core tunnel through a rectangular potential barrier. It is shown that resonance structures appear in the transmission spectrum in some cases, which reflects the existence of quasi-bound-states of the centre of mass motion around the potential barrier. It is also shown that an inelastic tunnelling occurs in which the transitions in the relative motion are induced by the tunnelling of the centre of mass.

Journal of Physics Condensed Matter•May 16, 1994•42 citations

Quantum tunnelling for thousands potential barriers

Domenico Oricchio

I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).

•Feb 20, 2026•0 citations

Time dependence and line shape of spontaneous quantum tunneling

B. U. Felderhof

The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.

Journal of Physics A Mathematical and Theoretical•Oct 7, 2008•2 citations

Resonance due to the interaction of tunneling particles with modulation quanta

Mark J. Hagmann

Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.

Applied Physics Letters•Feb 13, 1995•25 citations

Transmission coefficient in double barrier quantum tunnelling effect

Zhongzheng Duan, Wenxi Luo, Xiaohan Xu

At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.

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

Tunneling and transmission resonances of a Dirac particle by a double barrier

Vı́ctor M. Villalba, Luis A González-Árraga

We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.

arXiv:1001.4712•Physica Scripta•Feb 1, 2010•12 citations
Exact solutions are obtained for the diffusion in single- and square-well potentials, by using the Laplace transform method. We compare transmission of a classical particle through one barrier of given length and through two barriers with a well between them of the same overall length. It turns out that the existence of a well facilitates the transmission of a classical particle for intermediate times, which bears some resemblance to resonances tunneling through quantum barriers. \textcopyright{} 1996 The American Physical Society.
Some aspects of quantum tunnelling of a composite particle are clarified by a simple model. We calculate the probability that a couple of point particles bound to each other by a square well potential with a hard core tunnel through a rectangular potential barrier. It is shown that resonance structures appear in the transmission spectrum in some cases, which reflects the existence of quasi-bound-states of the centre of mass motion around the potential barrier. It is also shown that an inelastic tunnelling occurs in which the transitions in the relative motion are induced by the tunnelling of the centre of mass.
I use the Laplace transform to obtain the transmission coefficient for each number of potential barriers.There are a little number of close-form solution of Schröginger equation, and this is another exact solution.It is possible to obtain the tunnelling for electronic components (resonant-tunnelling diode), or neutronand proton diffusion in astrophysics (stellar nucleosyntheis) and nuclear physics (nuclear fission and fusion),or chemical tunnelling for two molecules reaction (that I solve in a generic case), or to obtain a quantumfilter for the energy spectrum (that I solve for some coating layers).
The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.