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:2507.06045 · Voprosy atomnoj nauki i tehniki. Seriâ âderno-fizičeskie issledovaniâ

QUANTUM TUNNELING IN THE TIME PERIODIC DOUBLE WELL POTENTIAL

Vladimir I. Dubinko, Oleksandr Serhiyovych Mazmanishvili

Published Dec 29, 2025• 0 citations
arXivHTMLPDF

Abstract

We present the mathematical model and numerical calculation results for the tunneling of the wave function in a time-periodic double-well potential. The biquadratic potential of a double-well form is used. Based on a mathematical model of the time evolution of the wave function, a numerical algorithm has been developed and a program has been created for solving the Schrödinger equation, which describes the time evolution of the particle's wave function. As a result of numerical modeling, modulation regimes have been obtained at which tunneling took place. In a stationary double well potential, no tunneling was observed. On the contrary, in the time periodic potential, tunneling was observed, the rate of which depended on the modulation frequency. Adjusting the modulation parameters, it is possible to control the particle wave function tunneling rate.

DOI
10.46813/2025-160-072

Related papers

Wave Function Penetration, Tunneling, and Quantum Wells

Uri Peskin

Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.

Cambridge University Press eBooks•May 11, 2023•0 citations

Quantum Dynamics in Canonical and Micro-canonical Ensembles. Part II. Tunneling in Double Well Potential

V S Filinov, Yu. E. Lozovik, A V Filinov +2 more

In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.

arXiv:quant-ph/9811002•Physica Scripta•Oct 1, 1998•3 citations

Effective tunneling coefficient of a coupled double-well system modulated by anharmonic periodic potentials

Noriaki Tsukada, Hiroyuki Yoshida, Takashi Suzuki

We numerically study coherent tunneling oscillations of the particles between two levels in a double-well potential in the presence of anharmonic periodic potentials. Extremely short driving pulses modify the tunneling coefficient to ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$, where $\ensuremath{\kappa}$ is the bare tunneling coefficient without the driving field and $A$ is the pulse area of the driving wave form. The modulation amplitude of the ${\ensuremath{\kappa}}_{\mathit{eff}}$ gradually decreases as the driving wave form becomes broad and is given by ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}{J}_{0}(A)$ for the sinusoidal modulation, where ${J}_{0}(x)$ denotes the ordinary Bessel function of order zero. Theoretical derivation of the effective tunneling coefficient ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$ is also shown for a periodic $\ensuremath{\delta}$ kick with alternating sign by means of the transfer matrix formula.

Physical Review A•Feb 1, 2008•4 citations

NUMERICAL CALCULATION OF QUANTUM TUNNELING BASED ON JULIA

Bingyi DU, Yan Xu

Quantum tunneling is an important model in elementary quantum mechanics. However, since solution of its wave function involves transcendental equation, many elementary quantum mechanics textbooks often focus on the explanation of transmission coefficient, and rarely mention the evolution of wave function. Although this process is mentioned in some textbooks, it is often solved by graphic method, which is not conducive to beginners' understanding of the process. In this paper, a numerical calculation method for beginners is proposed to solve this problem. According to Dirac's vectorization idea, the state vectors and operators in the Schrodinger equation are discretized in the form of vectors and matrices, respectively, and the time-dependent evolution of wave function tunneling under several potential barriers is solved by Julia numerically.

Wuli yu gongcheng.•Jun 1, 2022•0 citations

Tunneling control by external actions: Dynamic localization

В. М. Волохов, Sergey A. Tovstun

Tunnel dynamics described by the one-dimensional nonstationary Schrödinger equation with a potential periodically depending on time was studied numerically and analytically. Quasi-energy levels were found in the second order with respect to perturbation parameter powers. The dependence of the rate of tunneling on potential parameters was found numerically, and the results were analyzed using the two-level model. The condition of tunneling “quenching” for an arbitrary initial state was obtained.

Russian Journal of Physical Chemistry B•Jun 1, 2007•0 citations

Time evolutions of quantum mechanical states in a symmetric double-well potential

Peter Senn

The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.

American Journal of Physics•Mar 1, 1992•6 citations
We present the mathematical model and numerical calculation results for the tunneling of the wave function in a time-periodic double-well potential. The biquadratic potential of a double-well form is used. Based on a mathematical model of the time evolution of the wave function, a numerical algorithm has been developed and a program has been created for solving the Schrödinger equation, which describes the time evolution of the particle's wave function. As a result of numerical modeling, modulation regimes have been obtained at which tunneling took place. In a stationary double well potential, no tunneling was observed. On the contrary, in the time periodic potential, tunneling was observed, the rate of which depended on the modulation frequency. Adjusting the modulation parameters, it is possible to control the particle wave function tunneling rate.
Scanning tunneling microscopy is introduced as an analytic tool for characterizing nanostructures and dynamics on the nanoscale. To analyze the underlying tunneling phenomenon, the Schrödinger equation is solved for a particle confined between finite potential energy steps. For a single potential energy well, the proper solutions to the Schrödinger equation are similar to those of a particle in an infinite box, except for the emerging “tails” of the stationary wave functions in the “classically forbidden” regions of the external confining potentials. For a symmetric double well potential, wave function penetration of the stationary solutions manifests in their even distribution among the wells. An attempt to localize the particle in one of them would lead to periodic oscillations between the wells, as if the particle can “tunnel” under the separating energy barrier. The wave function penetration and tunneling phenomena are discussed also in relation to energy band formation in periodic lattices.
In the second part of this paper on micro-canonical ensemble a new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The time correlation functions have been presented in the form of the integral of the Weyl's symbol of the considered operators and the Fourier transform of the product of matrix elements of the dynamic propagators. For the latter function the integral Wigner–Liouville's type equation has been derived. The initial condition for this equation has been obtained in the form of the Fourier transform of the Wiener path integral representation of the matrix elements of the propagators at initial time. A numerical procedure for solving this equation combining both molecular dynamics and Monte Carlo methods has been developed. Numerical results have been obtained for series of average values of the quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunnelling particles at barrier crossing and the absorption spectra of electrons in a potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunnelling transitions and the effect of the quasi stationary state have been considered as the reasons for the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. The possibility of applying the developed approach to the theory of classical wave propagation in random media has also been considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides.
We numerically study coherent tunneling oscillations of the particles between two levels in a double-well potential in the presence of anharmonic periodic potentials. Extremely short driving pulses modify the tunneling coefficient to ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$, where $\ensuremath{\kappa}$ is the bare tunneling coefficient without the driving field and AAA is the pulse area of the driving wave form. The modulation amplitude of the ${\ensuremath{\kappa}}_{\mathit{eff}}$ gradually decreases as the driving wave form becomes broad and is given by ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}{J}_{0}(A)$ for the sinusoidal modulation, where J0(x){J}_{0}(x)J0​(x) denotes the ordinary Bessel function of order zero. Theoretical derivation of the effective tunneling coefficient ${\ensuremath{\kappa}}_{\mathit{eff}}=\ensuremath{\kappa}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}A$ is also shown for a periodic $\ensuremath{\delta}$ kick with alternating sign by means of the transfer matrix formula.
Quantum tunneling is an important model in elementary quantum mechanics. However, since solution of its wave function involves transcendental equation, many elementary quantum mechanics textbooks often focus on the explanation of transmission coefficient, and rarely mention the evolution of wave function. Although this process is mentioned in some textbooks, it is often solved by graphic method, which is not conducive to beginners' understanding of the process. In this paper, a numerical calculation method for beginners is proposed to solve this problem. According to Dirac's vectorization idea, the state vectors and operators in the Schrodinger equation are discretized in the form of vectors and matrices, respectively, and the time-dependent evolution of wave function tunneling under several potential barriers is solved by Julia numerically.
Tunnel dynamics described by the one-dimensional nonstationary Schrödinger equation with a potential periodically depending on time was studied numerically and analytically. Quasi-energy levels were found in the second order with respect to perturbation parameter powers. The dependence of the rate of tunneling on potential parameters was found numerically, and the results were analyzed using the two-level model. The condition of tunneling “quenching” for an arbitrary initial state was obtained.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.