Keywords
Elliptical geometry, Boundary conditions, Schrödinger equation, Superconductivity, Microstrip Antenna, Mathieu functions
Abstract
For a quantum particle confined to a two-dimensional elliptical box or electromagnetic wave in a microstrip antenna, geometrical and boundary condition interplay result in a spectrum of spatial patterns. Due to the asymmetrical nature of the ellipse, we are faced with continuous symmetry reductions, leaving both degenerate and nondegenerate solutions. Here, we present a complete derivation of an analytical solution and visualizations of the fundamental wavefunctions for both Dirichlet and Neumann boundary conditions respectively corresponding to the quantum elliptical box and the elliptical microstrip antenna.
We demonstrate that the eigenmodes, governed by eccentricity, directly correspond to the modal field distributions in the ellipse itself, with the fundamental modes exhibiting strong field confinement and efficient radiation. The symmetrical properties of the elliptical cavity influence the degeneracy and spatial structure of the wavefunction, with experimental consequences. Our results include color contour plots of representative wavefunctions, highlighting nodal structures and symmetry features unique to the elliptical geometry.
These findings provide a foundation for applications to optimize elliptical microstrip antennas in advanced applications, such as spacecraft and satellite communications, where tailored polarization and high-performance operation are required. Versatility extends to superconducting materials, including high-temperature Bi2Sr2CaCu2O8+d-based devices, enabling ultra-efficient, low-cost communications systems. Additionally, elliptical microstrip antennas support robust, broadband connectivity in efficient transportation systems such as bullet trains.
This abstract adapts the structure and core ideas from the provided example, emphasizing the role of symmetry, boundary conditions, and visualizations of wave functions, while specifically focusing on the elliptical geometry and its implications for quantum and antenna systems.
Thesis Completion Year
2025
Thesis Completion Semester
Fall
Thesis Chair
Klemm, Richard
College
College of Sciences
Department
Physics
Thesis Discipline
Physics
Language
English
Access Status
Open Access
Length of Campus Access
None
STARS Citation
Tikalal, Nishtha, "0th Order Solutions of the Wavefunctions for the Quantum Elliptical Box and Microstrip Antenna" (2025). Honors Undergraduate Theses. 465.
https://stars.library.ucf.edu/hut2024/465
Included in
Analysis Commons, Condensed Matter Physics Commons, Geometry and Topology Commons, Partial Differential Equations Commons, Quantum Physics Commons, Special Functions Commons
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