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A computational framework for multiply-connected and electromagnetic quantum systems
Dissertation   Open access

A computational framework for multiply-connected and electromagnetic quantum systems

Allyson O'Brien
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2017
DOI:
https://doi.org/10.17918/etd-7478
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Abstract

Finite element method Quantum theory Physics
In this dissertation, we develop the capabilities of the Finite Element Method (FEM) and Finite Element analysis (FEA) in the domain of computational quantum physics. We describe how FEM works and how it has been leveraged in quantum physics research over the last several decades. We derive new methods for modeling and analyzing quantum systems by using "holes" (cutouts) in the geometries of billiards in order to tune energy levels and energy level spacing. We address historical issues of the method in modeling systems with magnetic fields. These issues include non-convergence of gauge choice as well as non-convergence of solutions at higher energy levels. By developing a set of tools and a framework to form various 'admissible systems', we demonstrate that these issues stem from a misrepresentation of FEM algorithm design in quantum models. Through leveraging gauge-invariance in algorithm design, we describe how an appropriate unique gauge is identified for modeling various physical parameters. We then extend this idea into a framework that leverages various gauge selections in order to gain a much more complete picture of a quantum model and its various complementary observables. Finally, we show that this framework extends to modeling quantum systems that are bounded at realistically sized length-scales on the cusp of magnetic confinement. Through this work new limits on the canonical Dirichlet boundary conditions are defined.

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