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Journal of Inner Mongolia University for the Nationalities

Tunnel Effect of the Micro Particle

Cui-Lan Zhao

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

In this paper,transmission coefficient that the micro particle penetrate two potential barriers successively is calculated by solving the stationary schrodinger equation.Our numerical results show that the micro particle could penetrate potential barriers no matter energy of particle is taller or smaller than the height of potential barriers.But the transmission coefficient varies in several different circumstances.Particularly,it may arise resonance transmission in conditions.In a word,the transmission coefficient could changes with the height and width of two potential barriers,and have nothing to do with the distance between two potential barriers.

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Highlights in Science Engineering and Technology•Aug 20, 2024•0 citations
In this paper,transmission coefficient that the micro particle penetrate two potential barriers successively is calculated by solving the stationary schrodinger equation.Our numerical results show that the micro particle could penetrate potential barriers no matter energy of particle is taller or smaller than the height of potential barriers.But the transmission coefficient varies in several different circumstances.Particularly,it may arise resonance transmission in conditions.In a word,the transmission coefficient could changes with the height and width of two potential barriers,and have nothing to do with the distance between two potential barriers.
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 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.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.
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.
The quantum tunneling effect, which is based on the quantum theory that microscopic particles behave in waves through barriers higher than their own energy, has important applications in practical technology. Microscopes such as scanning tunneling microscopes, which use quantum tunneling to generate tunnel currents, have a resolution of up to the desired level. Moreover, it can be used in biology to observe molecular structures such as macromolecules and biofilms. In this paper, the transmission coefficients of one-dimensional asymmetric unilateral barrier and asymmetric two-sided barrier under general arbitrary boundary conditions are strictly derived using the conventional method of solving the steady state Schrodinger equation by quantum mechanics and matrix. Meanwhile, the transmission coefficients of symmetric two-sided barrier are numerically studied. From the above series of methods, it is deduced that the basic working principle of resonant tunneling diode and other semiconductor electronic devices is exactly the tunnel effect. Thus, it is found that quantum tunneling effect plays an important role in life.