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.
LI et al. (Sun,) studied this question.