Path integrals in non-relativistic quantum mechanics

Abstract

This thesis starts with a brief review of local and global approaches to classical mechanics. Path integral formalism, which constitutes a global approach to quantum mechanics, is introduced. A brief history of non-relativistic applications is given. The Lagrangian and the Hamiltonian forms of the path integral are derived. The path integral solutions of the simple problems of the free particle and the harmonic oscillator are given. The path integral solution of the fundamental problem of the hydrogen atom is reviewed. The Kustaanheimo-Stiefel transformation and the Duru-Kleinert time transformation are exhibited. It is shown that the Kustaanheimo-Stiefel transformation can be expressed in a simple form in terms of quaternions. A similar approach is applied to five dimensional hydrogen using octonions. A transformation between nine dimensional hydrogen and sixteen dimensional oscillator is derived as an example of higher dimensional generalizations. The path integral solutions to some problems that either have conceptual importance or demonstrate transformation techniques are given. These include: the free particle on a circle, the delta function potential, the P¨oschl-Teller potential, the Wood-Saxon potential, Rosen-Morse and Hulthen potentials, and the rigidly moving potential. The condition for the validity of the semi-classical approximation for the propagator in one dimension is derived.