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

College Physics

Dynamic simulation of potential barrier penetration based on the finite difference time omain method and Monte Carlo method

Yanan Zhang

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Both the wave function and the probability density of an electron wave during the process of barrier penetration are calculated based on the finite difference time domain method. Using the Monte Carlo method,we propose the probability density to be described by the denseness of random point and simulate dynamic process of potential barrier penetration.

Related papers

Computer Simulation of Dynamic Stochastic Process of Potential Barrier Penetration

Luo Li-jin

This paper demonstrates the dynamic and stochastic process of the potential barrier penetration of microcosmic particle,based on Monte Carlo method and the transmission coefficient of microcosmic particle potential barrier penetration,by using the programming and visualization of Matlab.

Journal of Shenyang Normal University•Jan 1, 2006•0 citations

The application of the Finite Difference Time Domain method in the problem of potential barrier

Qing Zhou

The application of the Finite Difference Time Domain(FDTD) method in quantum-mechanical problems becomes more and more interesting.The postulate,the boundary cut-off condition and the initial trial wave function are described.Based on the FDTD method,three different potential-barrier problems including rectangular potential-barrier,deltoid potential-barrier and syntonic potential-barrier are investigated.The energy variation and the probability variation are calculated and modeled,the important phenomenon of quantum tunnel effect reappeared.It is convenient to apply the FDTD method in quantum-mechanical problems and the results are acceptable while using the boundary cut-off based on the expectation energy of the system.And it is potentially further extended to other quantum-mechanical problems.

Journal of Yunnan Normal University•Jan 1, 2007•0 citations

The Finite-difference Time-domain Method and Its Application to Quantum Mechanics

Chijian Liao

This paper introduced the FDTD method,the basic principles of the boundary and initial treatment selection of the trial wave function.On this basis,the matrix calculation to simulate the barrier change the energy of the system were given,the probability of changes in the quantum tunneling effect reproduce the important physical phenomena.Further demonstrates the use of FDTD method to solve the problem of quantum mechanics is to facilitate workable and systematic approach to cut the average energy of the border can be more satisfactory results.

Journal of Gansu Lianhe University•Jan 1, 2010•0 citations

Numerical Solution of Quantum Tunneling

Qian‐Shu Li

The quantum hydrodynamics dynamics (QHD) is introduced to study the probability of electron through the tunneling junction. The complex form of the wave function was transformed into polar form, the time evolution equations of C and S were given in the Lagrange framework, and the explicit central differencing scheme was adopted for time integration of the QHD equations. In the Lagrangian framework, the grid points of the system will inevitable become scattered and unstructured during the evolution, the spatially smooth functions C and S are especially suitable for rational function approximation method to enable the implementation of a robust numerical scheme. As an example, the dynamics of a Gaussian wave packet is calculated and results are presented in the figures. In these figures, wave packet penetration of the barrier can be clearly observed, showing the quantum effect obviously.

Transactions of Beijing Institute of Technology•Jan 1, 2010•0 citations

Time-dependent Interactions in Tunnelling Dynamics

Luca Nanni

In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrödinger equation of the model is exactly solved. The probability density inside the barrier is calculated from the obtained wave function, proving that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their interference. Furthermore, the interference term is time-dependent and contribute to the scattering process duration. The tunnelling time is calculated as the time needed to get the probability density inside the barrier to zero. This is the minimum duration of the measurement process before detecting the particle beyond the barrier. Based on this, a new method of estimating the tunnelling time by energy experimental measuring is proposed.

arXiv:2203.04882•Revista Mexicana de Física•Feb 9, 2022•1 citation

The Penetration of a Potential Barrier by Electrons

Carl Eckart

A potential barrier of the kind studied by Fowler and others may be represented by the analytic function $V$ (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy $W$ is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and $W<{V}_{max}$.

Physical Review•Jun 1, 1930•2,463 citations
Both the wave function and the probability density of an electron wave during the process of barrier penetration are calculated based on the finite difference time domain method. Using the Monte Carlo method,we propose the probability density to be described by the denseness of random point and simulate dynamic process of potential barrier penetration.
The application of the Finite Difference Time Domain(FDTD) method in quantum-mechanical problems becomes more and more interesting.The postulate,the boundary cut-off condition and the initial trial wave function are described.Based on the FDTD method,three different potential-barrier problems including rectangular potential-barrier,deltoid potential-barrier and syntonic potential-barrier are investigated.The energy variation and the probability variation are calculated and modeled,the important phenomenon of quantum tunnel effect reappeared.It is convenient to apply the FDTD method in quantum-mechanical problems and the results are acceptable while using the boundary cut-off based on the expectation energy of the system.And it is potentially further extended to other quantum-mechanical problems.
This paper introduced the FDTD method,the basic principles of the boundary and initial treatment selection of the trial wave function.On this basis,the matrix calculation to simulate the barrier change the energy of the system were given,the probability of changes in the quantum tunneling effect reproduce the important physical phenomena.Further demonstrates the use of FDTD method to solve the problem of quantum mechanics is to facilitate workable and systematic approach to cut the average energy of the border can be more satisfactory results.
The quantum hydrodynamics dynamics (QHD) is introduced to study the probability of electron through the tunneling junction. The complex form of the wave function was transformed into polar form, the time evolution equations of C and S were given in the Lagrange framework, and the explicit central differencing scheme was adopted for time integration of the QHD equations. In the Lagrangian framework, the grid points of the system will inevitable become scattered and unstructured during the evolution, the spatially smooth functions C and S are especially suitable for rational function approximation method to enable the implementation of a robust numerical scheme. As an example, the dynamics of a Gaussian wave packet is calculated and results are presented in the figures. In these figures, wave packet penetration of the barrier can be clearly observed, showing the quantum effect obviously.
In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrödinger equation of the model is exactly solved. The probability density inside the barrier is calculated from the obtained wave function, proving that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their interference. Furthermore, the interference term is time-dependent and contribute to the scattering process duration. The tunnelling time is calculated as the time needed to get the probability density inside the barrier to zero. This is the minimum duration of the measurement process before detecting the particle beyond the barrier. Based on this, a new method of estimating the tunnelling time by energy experimental measuring is proposed.
A potential barrier of the kind studied by Fowler and others may be represented by the analytic function VVV (Eq. (1)). The Schr\"odinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy WWW is calculable (Eq. (15)). The approximate formula, $1\ensuremath{-}\ensuremath{\rho}=\mathrm{exp}{\ensuremath{-}\ensuremath{\int}\frac{4\ensuremath{\pi}}{h}{(2m(V\ensuremath{-}W))}^{\frac{1}{2}}\mathrm{dx}}$ is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W<VmaxW<{V}_{max}W<Vmax​.