Quantum mechanical problems in one, two and three dimensions

Abstract

<p>The underlying theme of this thesis is to solve mathematical models related to physical problems in one, two and three spatial dimensions. In some cases, particularly in one spatial dimension, the equations could be solved exactly, while for models in higher dimensions, the solutions were approximate. Although the four separate chapters of the thesis are largely independent, quantum statistics played an important role in the models considered. In the following paragraphs, I briefly summarize the contents of the four chapters. An exactly solvable field theoretical model in one spatial dimension and its classical bound state solution (including the zero mode) are presented in chapter one. The related bosonization and vacuum charge are discussed. In chapter two, a many-anyon system in two dimensional space with a confining potential is solved using the Thomas-Fermi mean-field method. The ground-state energy and spatial distribution are obtained as functions of the statistical parameter. In chapter three, Chern-Simions coupling to fermions in two dimensions is considered. A zero-mode soliton solution is obtained. Classical vortices in fluid mechanics are shown to be mathematically analogous to anyons. Some new results are derived using this analogy. Finally, in three-dimensional space, the vacuum properties of the Nambu-Jona-Lasinio model are studied at both zero and finite temperatures. The vacuum condensation energy per unit volume is calculated and is identified as the MIT bag constant. Some other thermodynamic properties are also calculated, showing striking similarity to the BCS superconductor. A scaling law of the chiral condensation energy density in a nuclear medium is suggested.</p>