Study of Scattering Caused by Repulsive Square Potential with Pocket Based on Maximum Amplitude Trajectories

Abstract

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.