Physics Master of Science Thesis Defense by Daniel Pierce
Abstract:
An electron in quantum confinement takes on a discrete energy spectrum which is defined based on the solution to the Schrödinger Equation for a given potential. Well defined closed-form energy spectra are known for the particle in a box, circular potential, quarter circle potential, and an equilateral triangle. A closed-form solution for more complex shapes may not be known, but numerical methods can be used to find an approximate solution. In this research, an application of the Finite Element Method (FEM) in Wolfram Mathematica is presented and applied to Quantum Billiards with a variety of geometries. To assess the accuracy of the method, the computed energy states are analyzed in the limit of a polygon with an increasing number of sides, and the numerical results are validated against analytical solutions for geometries with known exact forms. The FEM results closely match analytical solutions for known potentials, demonstrating its high accuracy. For high principal quantum number n, quantum scarring may emerge for certain geometries. The nature of quantum scarring and its presence in the computed models is also investigated.
Advisor: Dr. Renuka Rajapakse, Department of Physics
Committee members:
- Dr. David Kagan, Department of Physics
- Dr. Jay Wang, Department of Physics
Note: All PHY Graduate Students are encouraged to attend.
SENG 201
Dr. Renuka Rajapakse
rrajapakse@umassd.edu