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Journal of Physics Conference Series

WKB approximation for analyzing quantum tunneling effect through negative Kratzer potential

Herry Fridolin Lalus, H Yudhawardana, Nila Prasetya Aryani

Published Jun 1, 2021• 3 citations
Publisher

Abstract

Abstract Kratzer potential is a potential for bound states in molecular bonds, while negative Kratzer potential is a potential for scattering states which has a barrier form, and which clearly has properties opposite to the bound state potential. This type of potential is the main subject of this paper. When a non-relativistic particle which has lower energy moving towards the potential, then there is a probability for the particle to break through the potential. In this paper, we use the WKB (Wenztel-Kramers-Brillouin) method to analyze the physical conditions that must be met by the system in order for the tunneling processes to occur. We present these conditions in the form of the relationship among several quantum variables such as the mass of the particle, the energy of the particle, the maximum height of the barrier potential, equilibrium internuclear separation, and the enactment domain of the potential field effect on the particle when the particle breaks through that potential.

DOI
10.1088/1742-6596/1918/2/022026

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Abstract Kratzer potential is a potential for bound states in molecular bonds, while negative Kratzer potential is a potential for scattering states which has a barrier form, and which clearly has properties opposite to the bound state potential. This type of potential is the main subject of this paper. When a non-relativistic particle which has lower energy moving towards the potential, then there is a probability for the particle to break through the potential. In this paper, we use the WKB (Wenztel-Kramers-Brillouin) method to analyze the physical conditions that must be met by the system in order for the tunneling processes to occur. We present these conditions in the form of the relationship among several quantum variables such as the mass of the particle, the energy of the particle, the maximum height of the barrier potential, equilibrium internuclear separation, and the enactment domain of the potential field effect on the particle when the particle breaks through that potential.
We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.
Wenzel-Kramers-Brillouin (WKB) method being an approximation method in quantum mechanics has been applied widely. For the potential problems WKB approximation method is taken as an effective technology. With the help of the WKB approximation method the quantum conditions of bound States particle being in three different typical one dimension (1D) potential well models are derived in this paper. In addition, the idea of the WKB approximation method and the procedures dealing with the potential problems are demonstrated clearly, which are very helpful for understanding the quantum conditions of 1D potential well models deeply and mastering the WKB approximation method to solve some related problems in quantum mechanics.
The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.
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.
The usual WKB analysis for quantum tunneling applies when the tunneling action is large, as it is for tall, wide potential barriers. In contrast we analyze tunneling when the action is small, as it is for tunneling across a tall, thin barrier. We develop a perturbative analysis where the control parameter is the inverse of the area under the potential barrier and apply our technique to several examples in 1+11+11+1 dimensions. In resonant situations for bound particles we find that the tunneling probability grows with time as ∝t2\propto t^2∝t2, while in non-resonant situations it grows linearly with time. We evaluate not only the tunneling probability but also the time-dependent tunneling wavefunction for a particle that escapes to infinity, {\it i.e.} from a quasi-bound state to the continuum.
Quantum mechanics serves as the fundamental cornerstone for numerous areas of research. Finding simple and analytical quantum transmission coefficients for potentials is a rare occurrence. Notably, we have discovered a readily provable yet unreported symmetry tunneling law. For an infinite parabolic potential, it can be proved analytically that the sum of the one-dimensional tunneling probabilities at incident energies equal to the barrier height plus or minus any energy deviation is consistently 1. Based on this relationship, the original WKB approximation and the Kemble's formula can be generalized to the above-barrier energy region in a simplified but more accurate manner, referred to as the symmetric WKB approach in this context. For the realistic potential between two nuclei, numerical results demonstrate that the symmetry relationship at the above-barrier energy region and the sub-barrier energy is also well satisfied, as well as in the cases of the multichannel WKB method. The symmetry tunneling law has been effectively extended to the Eckart potential as well. Such symmetry in tunneling may have universal characteristics but its underlying reason remains to be uncovered, which may help us to further understand the quantum mechanisms.