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Minimum Barrier Height That Produces a Resonance

Syed Golam Rabbani

Published Jan 1, 1982• 0 citations
OpenAlex

Abstract

This thesis is concerned with a research project that aims to investigate the minimum barrier height that produces a sharp resonance by combining a square well potential with a square barrier potential. This combination is a simple quantum mechanical system. Yet, it deserves a thorough investigation. We deal with this elementary problem based on an advanced quantum mechanical technique i. e, analytically continuable perturbation theory. Before the systematic application of the analytically continuable perturbation theory, however, we need to orient ourselves to the entire project, determining the range of parameters in which thorough investigations must be carried out. A sufficient number of case studies are presented in this thesis. The solutions of the Schrodinger equation have been surveyed in a Wide range of ~meters with the help of the Sigma-9 computer. The behavior of phase shift and cross section as a function of energy has been studied for different combinations of the well and barrier. Finally by studying the shape of resonance peak, with three different values of barrier parameters, we have determined what kind of square well to be combined with each barrier in order to produce a typical case of sharp resonance.

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This thesis is concerned with a research project that aims to investigate the minimum barrier height that produces a sharp resonance by combining a square well potential with a square barrier potential. This combination is a simple quantum mechanical system. Yet, it deserves a thorough investigation. We deal with this elementary problem based on an advanced quantum mechanical technique i. e, analytically continuable perturbation theory. Before the systematic application of the analytically continuable perturbation theory, however, we need to orient ourselves to the entire project, determining the range of parameters in which thorough investigations must be carried out. A sufficient number of case studies are presented in this thesis. The solutions of the Schrodinger equation have been surveyed in a Wide range of ~meters with the help of the Sigma-9 computer. The behavior of phase shift and cross section as a function of energy has been studied for different combinations of the well and barrier. Finally by studying the shape of resonance peak, with three different values of barrier parameters, we have determined what kind of square well to be combined with each barrier in order to produce a typical case of sharp resonance.
This thesis is concerned with an elementary problem of quantum mechanics, and its purpose is to expand the use of the maximum amplitude trajectories. There are two motivations of this thesis: First, we want to investigate the change in scattering caused by a potential when its shape gradually changes. In particular, we are interested in understanding how a resonance phenomenon originates and in determining what is required of a potential to produce a sharp resonance. We start from a square repulsive potential which never produces a resonance. Subsequently, we carve a pocket in the inner region. As the pocket becomes deeper, we observe that a resonance is created and that it becomes ' sharper as the depth of the pocket increases. Secondly, we deal with this elementary quantum mechanical problem based on an advanced mathematical technique, i.e. the analytically continuable pertubation theory. That theory suggests that the method of the maximum-amplitude-trajectory (MAT) is useful. A substantial number of case studies are presented in this thesis. The solutions of the Schroedinger equation have been surveyed in a wide range of parameters with the help of the Vax-ll computer. The behavior of the phase shift as well as the cross section has also been studied for particular cases as a function of wave number. We compare the MAT determined for fifteen different sets of parameters of the potential. By the comparison of these fifteen results, we are able to understand systematically how the ratio and width parameters influence scattering. The results will be presented in the following order: 1) Introduction, 2) Review of scattering theory, 3) Review of Maximum Amplitude Trajectory, 4) Maximum Amplitude Trajectory for model potentials, 5) Results and discussions, 6) Conclusions.
This thesis is part of a research project that is investigating the effect of different potentials on the scattering of a particle which collides with the potentials. We all know that scattering theory has been an active research area for a long time and many improvements have been made by scientists. Several useful approximation theories are already seen in many physics text books. It is an important problem of quantum mechanics to handle scattering when the potential is strong. In particular, the resonances are interesting phenomena in scattering, and their causes should be systematically investigated. In this thesis we wish to avoid any approximation to investigate the potential and because of that we only deal with zero angular momentum. We also limit the potential to be a spherical symmetric square barrier with a pocket inside. The pockets are introduced to cause resonance when they become substantially large, and our results will be presented in a series of graph and their meaning will be discussed.
Two rectangular models described by the one-dimensional Schr\"odinger equation with sharply localized potentials are suggested. The potentials have a multilayer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structures behave as a perfectly reflecting wall. The transcendental equations with respect to potential strengths, the solutions of which determine transmission (resonance) sets, are derived. In this regard, both the models are exactly solvable. The energy dependence of an incident particle is shown to reveal a resonance behavior, being completely different from that observed in a typical double-barrier structure.
The restrictions of micro-scale systems are approaching rapidly. In anticipation of this development, nano-scale electronics has become the focus of many researchers and engineers in academia and industry since early 1990s. The basic building blocks of modern integrated circuits have been diodes and transistors with their current-voltage I-V characteristics being of prime significance for the design of complex signal processing and shaping devices and systems. Classical and semi-classical physical principles are no longer powerful enough or even valid to describe the phenomena involved. The application of rich and powerful concepts in quantum theory has become indispensable. These facts have been influential in undertaking this research project. This research is built upon the determination of the Eigenpairs of one and two dimensional positive differential operators with periodic boundary conditions. The Schrödinger equation was solved for positive operators in both one and two dimensions. Fourier series were used to express the derivatives as the summation of Fourier terms. This led to a novel approach for the calculation of the eigenmodels of a perturbed potential well. The perturbation can be done via an electric field applied to the potential well. The research in this thesis includes a thorough understanding of quantum mechanics fundamentals, mastering of different approximation techniques such as the variational technique and results that have been generated and published using the novel techniques.
Abstract Quantum tunneling through potential barriers plays a central role in diverse physical phenomena ranging from nuclear fusion and radioactive decay to semiconductor devices and superconducting technologies. In this work, we calculate tunneling probabilities and wave functions for an arbitrary number of potential barriers with diverse functional forms, including square, Gaussian, parabolic, and Lorentzian profiles, using a general numerical framework. The numerical results are first validated by comparison with exact analytical solutions for double, triple and quadruple-square barrier systems, demonstrating excellent agreement. The results reveal that for a system with N barriers, a single resonance peak splits into ( N − 1 ) distinct peaks, which eventually merge into an energy band as N becomes large, in a manner fully consistent with the existing literature. A key additional result is the incorporation of absorption effects through an imaginary component of the potential, providing a realistic description of particle trapping and leading to a progressive suppression of resonant transmission. For sufficiently strong absorption, the resonant structures are completely washed out. Further, if the barriers become non-uniform or disordered, the regular resonance pattern is also observed to be completely washed out. These findings demonstrate the broad applicability of the approach to complex potential landscapes and offer insights relevant to both fundamental quantum mechanics and the design of next-generation quantum devices.