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Highlights in Science Engineering and Technology

Simulation Analysis of Factors Affecting the Quantum Tunneling Effect in 1D Double Potential Barriers Structures

Chenyang Liu

Published Jun 11, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.54097/vyzsp664

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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.
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.
The comprehension of the tunnel effect, the passage of matter waves and particles across a high potential barrier, was the most astounding of all discoveries in quantum mechanics. Research on tunnelling in semiconductors and superconductors, as well as the development of scanning tunnelling microscopy, garnered five Nobel prizes in physics. In this study, the resonant tunnelling effect of electron tunnelling through a one-dimensional double-potential barrier was found and verified by solving the time-independent Schrödinger equation and calculating the transmittance rate of the electron tunnelling through the two potential barriers by using the transmission matrix technique performed by MATLAB. According to the analysis, the transmittance rate was one under certain specific conditions (under some specific incident energy values or some specific distance between two potential barriers), which was called as resonant tunnelling. However, in this study, square potential wells were analyzed, which can be considered merely as ideal models in real life, and therefore all results are idealized. For further investigations, researchers can use more realistic parameters to set up potential wells and analyze real-life problems.
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.
Quantum theory plays a significant part in the modern physics and makes modern technology rapidly developed. Quantum tunnel effect is one of the most important discoveries. The exact expression of potential barrier is tough to derive, even for the double barrier. Many researchers have analysed the applications of quantum tunnel effect in real electronic devices. However, the exact expression of double barrier’s transmission coefficient was discussed rarely. In this article, the exact expression of the transmission coefficient will be given and the effect of parameters of barrier such as the width and the height will be discussed. In addition, the difference between symmetric and asymmetric double barrier will be given. This paper has the discussion of the transmission coefficient in order to not only give a clear understanding of the effect of parameters of double barrier to tyro and researchers, but also give a simpler way to consider the periodic barrier.
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.
Quantum tunneling is conventionally described using the stationary Schrödinger equation with an externally imposed potential barrier. In this work, we present a dynamical formulation of quantum tunneling in which the barrier is modeled as a structured medium possessing quantized internal modes that interact coherently with an incident electron. Using the Heisenberg operator formalism and a second-quantized representation of the barrier medium, we derive coupled dynamical equations governing the electron–barrier interaction. Under a continuum-mode and mean-field approximation, the collective response of the barrier modes generates an effective potential that reproduces the conventional rectangular barrier model and the standard tunneling transmission probability obtained from the Schrödinger equation. Within this framework, a tunneling traversal time is naturally defined from the dynamical evolution of the electron and is shown to depend on the barrier width, barrier height, and incident electron energy. Numerical simulations illustrating the transmission probability and tunneling-time behavior are presented. The results provide a complementary microscopic interpretation of tunneling processes in structured quantum media and may be relevant to nanoscale transport, photonic barriers, and coherent quantum devices.