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arXiv:2008.12957

On the quantum tunneling of one dimensional coulomb singular potential barrier

Atom Zh. Muradyan, G. A. Muradyan

Published Aug 29, 2020• 1 citation
arXivHTMLPDF

Abstract

How quantum tunneling will behave when the singularity is preserved as much as possible is the main question of this paper. We get that the Coulomb sibgularity is reflected as infinitly accelerated oscillations in the transmission coefficient between zero and one when the incident particles energy approaches the zero boundary. At relatively high energies, the tunneling acqures the caracter inherent in regular potentials, and becomes completly transparent in the asymptote.

DOI
10.48550/arxiv.2008.12957

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How quantum tunneling will behave when the singularity is preserved as much as possible is the main question of this paper. We get that the Coulomb sibgularity is reflected as infinitly accelerated oscillations in the transmission coefficient between zero and one when the incident particles energy approaches the zero boundary. At relatively high energies, the tunneling acqures the caracter inherent in regular potentials, and becomes completly transparent in the asymptote.
Abstract To get around the difficulty that the singular nature of the potential function V(x) = V0/|x|α introduces into physics, the regularization methods are used. However, they affect the singular nature of the problem, and so we discuss here how quantum tunneling behaves if the original singular nature of the Schrodinger equation remains unperturbed. For this purpose, I am starting from the precondition that the singular terms are mutually compensated in the current probability density and the current can be considered continuous. As a result, it is obtained that the mild-singular potential (with 0 < α < 1) has finite but unusual tunnel transparency, in particular, a non-zero value at zero energy of the incident particle. The transparency of a Coulomb potential well oscillates infinitely at zero energy, and only the strongly singular potential (with α > 1) repeats the property of the regularized prototypes: to be impenetrable.
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 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.
The problem of quantum tunneling through the singular potential barrier $$V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.$$ is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.
Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction     x n  .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1  , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0  , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.