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Journal of Physics Conference Series

Approximation Rectangular Function as Potential Barrier

Iftika Philo Wardani, Nufida Dwi Aisyah, Adri Supardi

Published Jan 1, 2020• 1 citation
Publisher

Abstract

Abstract Quantum tunneling occurs in the isomerization of hydroxymethylene into formaldehyde. Experiments and theory have proven the occurrence of it but both of them have a different result of the half-life of 0.7 hours. The differences the result is the motivation in this study. In theoretical, the probability of quantum tunneling calculates by Wentzel, Kreamer, Berlioun (WKB) approximation. This study calculates probability quantum tunneling use WKB and rectangular function approximation. The calculation of quantum tunneling probability with rectangular function approximation started at 1, 3, 5, 7 until 59 rectangular functions potential barrier use transfer matrix method to simplify the calculation. We use Gaussian function as the potential barrier that the height and width of the potential barrier satisfy the character of the potential barrier in isomerization hydroxymethylene barrier to formaldehyde. And the result, we get the minimum number of rectangular potential barrier to obtain a stable quantum tunneling probability value of 35 potential.

DOI
10.1088/1742-6596/1445/1/012009

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Abstract Quantum tunneling occurs in the isomerization of hydroxymethylene into formaldehyde. Experiments and theory have proven the occurrence of it but both of them have a different result of the half-life of 0.7 hours. The differences the result is the motivation in this study. In theoretical, the probability of quantum tunneling calculates by Wentzel, Kreamer, Berlioun (WKB) approximation. This study calculates probability quantum tunneling use WKB and rectangular function approximation. The calculation of quantum tunneling probability with rectangular function approximation started at 1, 3, 5, 7 until 59 rectangular functions potential barrier use transfer matrix method to simplify the calculation. We use Gaussian function as the potential barrier that the height and width of the potential barrier satisfy the character of the potential barrier in isomerization hydroxymethylene barrier to formaldehyde. And the result, we get the minimum number of rectangular potential barrier to obtain a stable quantum tunneling probability value of 35 potential.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.
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.
Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.
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.
Proton transfer plays a key role in the applications of advanced energy materials as well as in the functionalities of biological systems. In this work, based on the transfer matrix method, we study the quantum effects of proton transfer in a series of one-dimensional (1D) model potentials and numerically calculate the quantum probability of transferring across single and double barriers (wells). In the case of single barriers, when the incident energies of protons are above the barrier height, the quantum oscillations in the transmission coefficients depend on the geometric shape of the barriers. It is found that atomic resonant tunneling (ART) not only presents in the rectangular single well and rectangular double barriers as expected, but also exists in the other types of potential wells and double barriers. For hetero-structured double barriers, there is no resonant tunneling in the classical forbidden zone, i.e., in the case when the incident energy (E i) is lower than the barrier height (E b). Furthermore, we have provided generalized analysis on the characteristics of transmission coefficients of hetero-structured rectangular double barriers.