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Systematic Survey of Scattering by Repulsive Square Potentials with Pocket

B. Zhou

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

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.

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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.
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 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.
The purpose of this investigation is to contribute to the theory of one-dimensional scattering through a thorough study of the two-step potential. For this purpose we have studied the effect of changing a simple square well potential near its edge by adding a second step. The problem is treated using both perturbational and non-perturbational methods. The order of procedure is: (1) a general introduction, (2) multiple reflections of a wave packet by an attractive potential well, (3) discussion of the- two-step potential using perturbation theory, (4) calculation of depth of potential well for zero energy resonance, (5) conclusion.
The most important parameters in the study of low-energy scattering are the s-wave and p-wave scattering lengths and the s-wave effective range. We solve the scattering problem and find two useful formulas for the scattering length and the effective range for any angular momentum, as long as the Wigner threshold law holds. Using that formalism, we obtain a set of useful formulas for the angular-momentum scattering parameters of four different model potentials: hard-sphere, soft-sphere, spherical well, and well-barrier potentials. The behavior of the scattering parameters close to Feshbach resonances is also analyzed. Our derivations can be useful as hands-on activities for learning scattering theory.
This thesis is concerned with momentum anisotropy in strongly correlated electron systems, and explores its origin and its consequences through two contrasting projects. The first is a study of the temperature dependences of magnetotransport quantities in the normal state of the cuprate high-temperature superconductors. A phenomenological anisotropic small-angle scattering model is investigated; Hall effect measurements can be reproduced for parameters sufficiently close to particle-hole symmetry, but the experimentally observed magnetoresistance cannot be explained. The second project studies the phase diagram and quasiparticle properties of the square lattice Hubbard model within two-site cluster dynamical mean field theory (DMFT), at zero temperature. The "two-site" approach provides a drastically simplified but physically motivated self-consistency scheme for DMFT. This is combined for the first time with cluster DMFT, within which different magnetic orders and momentum anisotropy may be represented consistently. The extent of antiferromagnetism is determined; phases are discovered where the Fermi surface consists of small hole pockets, and the Mott transition happens as these pockets shrink to points. Anisotropic phenomena observed in the cuprates are reproduced by the theory; a pseudogap destroys the Fermi surface in some places, leaving behind Fermi arcs that closed into hole pockets by lines with very small quasiparticle residue.