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

International Journal of Quantum Chemistry

Tunneling phenomena in three‐dimensional double‐well potentials

Ruben G. Carbonell, Morton D. Kostin

Published Mar 1, 1973• 8 citations
Publisher

Abstract

Abstract The tunneling of a hydrogen atom through the barrier of a three‐dimensional double‐well potential is considered. From the time‐dependent Schrödinger equation, expressions are derived for the ensemble‐averaged probability density and for the probability that the hydrogen atom is in the reactant region, in the barrier region, or in the product region. It is found that when thermal vibrations are not taken into account, the ensemble‐averaged probability density may oscillate with time about its equilibrium value. When thermal vibrations are included, the oscillations become damped and the probability density approaches equilibrium. The tunneling rate is found to decrease considerably for increasing barrier thickness and barrier height.

DOI
10.1002/qua.560070214

Related papers

Tunneling dynamics of a double‐well oscillator: Effects of barrier fluctuation and thermal modulation

Aparna Saha, Pranab Sarkar

Abstract The tunneling dynamics of a particle moving in a bistable potential with fluctuating barrier is studied. For barriers fluctuating randomly in time we show by exact numerical calculation the significant effect of barrier fluctuation on the tunneling behavior of the particle. At nonzero temperatures the computed tunneling rate constant passes through a maximum when plotted against fluctuation frequency. The resonant frequency (at which the maximum appears) slowly decreases with increase in temperature and attains a constant value at higher temperature and it increases linearly with increase in barrier height of the potential. Another important observation is that in presence of barrier fluctuation the dependence of tunneling rate constant on temperature is strongly guided by the barrier fluctuation frequency. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 280–285, 2003

International Journal of Quantum Chemistry•Jan 1, 2003•5 citations

Simulation Analysis of Factors Affecting the Quantum Tunneling Effect in 1D Double Potential Barriers Structures

Chenyang Liu

The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.

Highlights in Science Engineering and Technology•Jun 11, 2024•0 citations

Transient tunneling effects of resonance doublets in triple barrier systems

Roberto Romo, Jorge Villavicencio, Gastón García‐Calderón

Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.

arXiv:quant-ph/0210009•Physical review. B, Condensed matter•Jul 16, 2002•20 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

Tunnelling in a double-well system with allowance made for the relaxation

Michael V. Basilevsky, Victor M. Ryaboy

Tunnelling in a one-dimensional double-well system with five turning points can be considered as one of the simplest models for a unimolecular chemical reaction. It is assumed that a potential barrier separating reactants from products is sufficiently high, so that the reaction rate constant decreases exponentially. The height of the second potential barrier is related to the widths of the energy levels in a product well imitating relaxation processes of reaction products. An analytical expression is derived for the width of an initial energy level in a reactant well. It depends exponentially on the value of 2W 1 (W 1 is a quasiclassical phase integral under the potential barrier separating reactants from products). The preexponential factor is expressed in terms of matrix elements of the propagator connecting solutions of the Schrödinger equation in asymptotic regions. Quantum and quasiclassical calculations of the propagator were performed. Under resonance conditions, when reactant and product energy levels coincide, the width of a reactant level is shown to be 4 exp 2W 2 times as large as its value in the absence of the resonance (W 2 is a quasiclassical phase integral under the second potential barrier).

Molecular Physics•Nov 1, 1981•6 citations

Resonances and oscillations in tunneling in a time-dependent potential

M. Ya. Azbel

A quantum particle, tunneling in a stationary potential, passes through a region with a weak slowly changing potential. Its probability density has resonances and oscillations in energy and in time, both in the barrier and beyond it. This is true in any dimensionality, for any type of waves (for example, de Broglie, electromagnetic, acoustic, and hydrodynamic waves).

Physical review. B, Condensed matter•Mar 15, 1991•25 citations
Abstract The tunneling of a hydrogen atom through the barrier of a three‐dimensional double‐well potential is considered. From the time‐dependent Schrödinger equation, expressions are derived for the ensemble‐averaged probability density and for the probability that the hydrogen atom is in the reactant region, in the barrier region, or in the product region. It is found that when thermal vibrations are not taken into account, the ensemble‐averaged probability density may oscillate with time about its equilibrium value. When thermal vibrations are included, the oscillations become damped and the probability density approaches equilibrium. The tunneling rate is found to decrease considerably for increasing barrier thickness and barrier height.
Abstract The tunneling dynamics of a particle moving in a bistable potential with fluctuating barrier is studied. For barriers fluctuating randomly in time we show by exact numerical calculation the significant effect of barrier fluctuation on the tunneling behavior of the particle. At nonzero temperatures the computed tunneling rate constant passes through a maximum when plotted against fluctuation frequency. The resonant frequency (at which the maximum appears) slowly decreases with increase in temperature and attains a constant value at higher temperature and it increases linearly with increase in barrier height of the potential. Another important observation is that in presence of barrier fluctuation the dependence of tunneling rate constant on temperature is strongly guided by the barrier fluctuation frequency. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 280–285, 2003
The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.
Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.
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.
Tunnelling in a one-dimensional double-well system with five turning points can be considered as one of the simplest models for a unimolecular chemical reaction. It is assumed that a potential barrier separating reactants from products is sufficiently high, so that the reaction rate constant decreases exponentially. The height of the second potential barrier is related to the widths of the energy levels in a product well imitating relaxation processes of reaction products. An analytical expression is derived for the width of an initial energy level in a reactant well. It depends exponentially on the value of 2W 1 (W 1 is a quasiclassical phase integral under the potential barrier separating reactants from products). The preexponential factor is expressed in terms of matrix elements of the propagator connecting solutions of the Schrödinger equation in asymptotic regions. Quantum and quasiclassical calculations of the propagator were performed. Under resonance conditions, when reactant and product energy levels coincide, the width of a reactant level is shown to be 4 exp 2W 2 times as large as its value in the absence of the resonance (W 2 is a quasiclassical phase integral under the second potential barrier).