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Wuli yu gongcheng.

NUMERICAL SIMULATION OF TIME-DEPENDENT EVOLUTION OF QUANTUM TUNNELING BASED ON SPLITTING OPERATOR METHOD

Haifeng LI, Sizhe HUANG, Zhe Liu, Leilei PAN

Published Feb 1, 2026• 0 citations
Publisher

Abstract

In this paper, the dynamic evolution process of Gaussian wave packet which describes a microscopic particle in one-dimensional and two-dimensional potential barriers are numerically studied by virtue of the split operator method. Moreover, the expected values of coordinates, momentum and Hamiltonian operators of the particle are calculated, and the Heisenberg uncertainty relationship is verified throughout the quantum tunneling process. Specifically, the time evolution operator is split into two parts which independently include the contributions of kinetic energy and potential energy. The wave function between coordinate and momentum representations is converted arbitrarily by utilizing the fast Fourier transform, and the numerical solution of the time-dependent Schrödinger equation is achieved. Accordingly, the results indicate that if the potential barrier is time independent, the time evolution of average total energy of the system is conserved on a large scale, but not strictly conserved on a small scale. When microscopic particle tunnels through a potential barrier, the total energy fluctuates rapidly. While leaving the potential barrier, it restores to the total energy value before quantum tunneling. Furthermore, in a one-dimensional double well potential, Gaussian wave packet oscillates back and forth within the double potential well, frequently crossing the middle potential barrier. Then the evolution of average energy oscillates rapidly and periodically with small amplitude. For the two-dimensional potential barrier, the reflection direction of the wave packet may not be the same as the incident direction, and after a period of time, particle mainly appears on both sides of the potential barrier. As a consequence, this paper presents a numerical foundation for the visualization research of quantum mechanics teaching and the in-depth exploration of quantum tunneling dynamics, deepening the understanding of wave-particle duality for the microscopic particle and Heisenberg's uncertainty principle.

DOI
10.26599/phys.2026.9320116

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In this paper, the dynamic evolution process of Gaussian wave packet which describes a microscopic particle in one-dimensional and two-dimensional potential barriers are numerically studied by virtue of the split operator method. Moreover, the expected values of coordinates, momentum and Hamiltonian operators of the particle are calculated, and the Heisenberg uncertainty relationship is verified throughout the quantum tunneling process. Specifically, the time evolution operator is split into two parts which independently include the contributions of kinetic energy and potential energy. The wave function between coordinate and momentum representations is converted arbitrarily by utilizing the fast Fourier transform, and the numerical solution of the time-dependent Schrödinger equation is achieved. Accordingly, the results indicate that if the potential barrier is time independent, the time evolution of average total energy of the system is conserved on a large scale, but not strictly conserved on a small scale. When microscopic particle tunnels through a potential barrier, the total energy fluctuates rapidly. While leaving the potential barrier, it restores to the total energy value before quantum tunneling. Furthermore, in a one-dimensional double well potential, Gaussian wave packet oscillates back and forth within the double potential well, frequently crossing the middle potential barrier. Then the evolution of average energy oscillates rapidly and periodically with small amplitude. For the two-dimensional potential barrier, the reflection direction of the wave packet may not be the same as the incident direction, and after a period of time, particle mainly appears on both sides of the potential barrier. As a consequence, this paper presents a numerical foundation for the visualization research of quantum mechanics teaching and the in-depth exploration of quantum tunneling dynamics, deepening the understanding of wave-particle duality for the microscopic particle and Heisenberg's uncertainty principle.
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.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
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.
Quantum wave dynamics in systems with potential wells, bathers and outer impenetrable walls has been studied using solutions of time-dependent Schrodinger equation. Numerical integration of Schrodinger equation, calculation of dynamical variables and its average values, Fourier-spectra were carried out for different initial conditions and different relations between wave packet energy and bather height. In non-symmetric wells the oscillations of wave packets are accompanied by multistep discrete tunneling of probability fluid from well through barrier. In double well potential (symmetric wells) the temporal scales of oscillations and tunneling were investigated in details. Influence of additional time-dependent potential on quantum wave packet dynamics in above listed systems is also discussed.
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.
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.