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SEMICLASSICAL TUNNELING EFFECT

Johan Öhman

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.

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This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
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.
This paper focuses on the quantum tunneling effect, which describes how a particle may penetrate barriers or obstructions using the Schrödinger equation. If a particle makes a barrier tunneling, it means that a particle with low energy goes through the barrier whose energy level is much higher. It is a basis or background for the entire theory of this part. The particle transport probability is explained using both single- and multi-barrier system models. The paper studies the relationship between barrier parameters and particle transmission probability for both single- and multiple-barrier system models.The aim of the study is the provision of the theoretical devices and algorithms via a quantum operational efficiency.Particularly, the paper deals with the multipath barrier tunneling effect, and it connects the transmission coefficients with the system parameters. The goal of this article is to come up with the new strategies and procedures that one can use the quantum phenomenon of the tunneling effect in the contemporary physical th applications, cutting the gap between theoretical physics and technology.This will be done by studying and analyzing theoretical processes and mathematics of this phenomenon along with the technological applications it has.
The effects of coupling to a harmonic oscillator on the quantum tunneling of a macroscopic motion are studied through the influence functional formalism of Feynman's path integral method for the general coupling form factor. As an example, we consider the model in which the potential barrier is parabolic and the coupling Hamiltonian is linear in both coordinates of the macroscopic motion and of the intrinsic harmonic oscillator. The results are then compared with the exact solution obtained through the canonical transformation into normal coordinates in the limiting cases when the normal coordinates reduce to the original coordinates. We found that: (1) In the adiabatic case, i.e., when the recurrence time $\frac{\ensuremath{\pi}}{\ensuremath{\omega}}$ of the oscillator is much shorter than the transmission time through the macroscopic potential barrier, the effect of oscillator coupling can be well represented by an effective potential. The coupling enhances the tunneling probability on the whole. (2) There exists a critical energy, above which the tunneling probability is reduced because of the linear oscillator coupling. In the weak coupling limit and when $\ensuremath{\omega}\ensuremath{\rightarrow}0$, the critical energy becomes $\ensuremath{-}\ensuremath{\infty}$, so that the coupling to the oscillator always reduces the tunneling probability.NUCLEAR REACTIONS Quantum tunneling, coupling of macroscopic motion to intrinsic oscillators, semiclassical method, heavy ion fusion.
We show how quantum mechanical barrier reflection and transmission coefficients ℛ and 𝒯 can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V ( x ) = ∑ j V ( j ) ( x ), where each V ( j ) goes to zero for x →±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual ℛ ( j ) and 𝒯 ( j ) . The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.
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.