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Open Access Journal of Science

Tunneling transmission coefficient from the external electric field barrier

Hasan Hüseyin Erbil

Published Sep 7, 2020• 1 citation
Publisher

Abstract

According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is well known. This event is called tunneling. Tunneling is a probability, and it is measured by a transmission coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In previous publications, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values. In this article, this general transmission coefficient is applied to calculate transmission coefficient for an arbitrary form external electrical field barrier (cold emission by the metal). The values obtained from our calculations and those of classical method (WKB) have been compared.

DOI
10.15406/oajs.2020.04.00159

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According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is well known. This event is called tunneling. Tunneling is a probability, and it is measured by a transmission coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In previous publications, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values. In this article, this general transmission coefficient is applied to calculate transmission coefficient for an arbitrary form external electrical field barrier (cold emission by the metal). The values obtained from our calculations and those of classical method (WKB) have been compared.
According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by a transition coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In this paper, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values.
According to theories of classical physics, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by transition coefficient. Correct calculation of this coefficient is particularly important because extremely sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient was generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In the previous papers, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results were applied to calculate the half-life values of the nuclei that emit alpha particle. In this article, the half-life calculation formulas obtained in previous studies were made simpler. It has been shown that it is applicable to all radioactive decays (alpha, beta, gamma etc.). The half-life values obtained from our calculations have been compared with the experimental values, and it has been found that the new half-life values are exactly consistent with the experimental values.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
A simple, yet accurate, method of solving the Schrödinger equation across an arbitrary potential barrier is described, based on the analytic transfer-matrix technique. This paper gives the final analytical expression for the transmission probability. Numerical calculations for four typical potential barrier structures are compared with the traditional WKB, modified WKB and the modified Airy function (MAF) methods. Excellent agreement with an exact example is demonstrated. Most importantly, in cases where other potential-barrier methods fail, our method performs well.
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.
A simle numerical method of calculating transmission coefficients across arbitrary potentials is presented by using the combination of a transfer matrix procedure and the solution of the wave functions in arbitrary potentials.This method is applied to a parabolic potential and a double barrier potential in an electric field.The transmission coefficients calculated are in agreement with those obtained using a multistep potential approximation.The wave functions at the resonant states are also given for the two examples.