The article presents calculations and visualization of wave packet tunneling through the potentialbarrier of various widths done by using the MATLAB software. The graph of distribution of the probability densityalong barrier width is obtained in two and three dimensions. The article contains materials about quantummechanical tunnel effect such as the reflection of the part of the wave packet from the potential barrier and passingthrough the barrier of the other part. Calculations and visualization of the tunneling process are performed forvarious widths of the barrier. Figures given in the article show the probability of finding the particle at a certain time,i.e. the square of the absolute value of the state vector 2 . The part of the total number of particles tunnels throughbarrier, the other part reflects from it. The probability of tunneling increases with decrease of the barrier width. Thedetailed analysis of the graphs reveals that the wave function exponentially decreases inside the potential barriertherefore the observation of this effect requires a rather small width of the barrier. Students are offered to developprograms for various widths of the potential barrier. Results of experiments are used during the study of the quantummechanics.
The article presents calculations and visualization of wave packet tunneling through the potentialbarrier of various widths done by using the MATLAB software. The graph of distribution of the probability densityalong barrier width is obtained in two and three dimensions. The article contains materials about quantummechanical tunnel effect such as the reflection of the part of the wave packet from the potential barrier and passingthrough the barrier of the other part. Calculations and visualization of the tunneling process are performed forvarious widths of the barrier. Figures given in the article show the probability of finding the particle at a certain time,i.e. the square of the absolute value of the state vector 2 . The part of the total number of particles tunnels throughbarrier, the other part reflects from it. The probability of tunneling increases with decrease of the barrier width. Thedetailed analysis of the graphs reveals that the wave function exponentially decreases inside the potential barriertherefore the observation of this effect requires a rather small width of the barrier. Students are offered to developprograms for various widths of the potential barrier. Results of experiments are used during the study of the quantummechanics.
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.
Quantum tunneling is a phenomenon where particles have a probability of penetrating a potential barrier despite having total energy llower than the barrier height. This study analyzes the tunneling effect by solving the one-dimensional (1D) Schrodinger equation using the finite difference method to obtain the wave function evolution for various potential barrier configurations. The solution is implemented in a Graphical User Interface (GUI) MATLAB to facilitate analysis and visualization, allowing users to interactively adjust potential parameters, energy, and other conditions. Simualation results demonstratehow transmission probability depends on energy, height and width of potential barrier. This GUI provides an intiuitive tool for exploring quantun tunneling, making it valuable for both education and research in quantum physics.
The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space.
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.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
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.