Mahurin Honors College Capstone Experience/Thesis Projects

Department

Physics and Astronomy

Additional Departmental Affiliation

Mathematics

Document Type

Thesis

Abstract

This work explores the development and unification of quantum theory within phase space, with a particular focus on new mathematical structures and solution techniques that extend the standard quantum formalism. Building upon the Quantum Phase Space Representation (QPSR) introduced by Torres-Vega and Frederick, we present several advancements that address longstanding gaps in the formulation and solvability of quantum systems in this framework. We introduce the Half-Transform Ansatz, a novel method for solving the Time-Independent Schrodinger Equation by recasting it into a hypergeometric form, enabling the application of the Nikiforov-Uvarov method. This approach is demonstrated through the analysis of quarkonium systems using the Cornell potential, yielding wave functions in phase space and energy spectra consistent with experimental and theoretical models. The methods is shown to generalize to a broad class of two-particle systems with scleronomic potentials. We also introduce the Operator Space Manifold Theory, a geometric model in which quantum operators are treated as points on a Riemannian manifold. This idea offers a foundation to understanding how exponential shifts result in new operator representations for quantum phase space.

Extending beyond specific systems, we reformulate quantum mechanics on the cotangent bundle, employing nonlinear connections to geometrically partition phase space into position and momentum submanifolds. This leads to a modified Fourier transform, modified bra-ket notation, considerations for a metric operator, and a geometric quantization procedure that constructs quantum operations from classical observables via Hamiltonian flows. Global quantum operations then act on sections of a complex line bundle whose existence is proven using Weil's integrality condition. This establishes the quantizability of Hamilton spaces and provides a rigorous framework for quantum theory in curved phase spaces.

Together, these contribution provide a comprehensive framework that connects operator theory, geometry, and phase space representations - laying the found work for a unified and generalizable quantum theory in both flat and curved settings.

Advisor(s) or Committee Chair

Tony Simpao, Ph.D.

Disciplines

Mathematics | Physics

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