Committee Chair

Belinskiy, Boris

Committee Member

Cox, Christopher L.; Kong, Lingju; Nichols, Roger

Department

Dept. of Mathematics

College

College of Arts and Sciences

Publisher

University of Tennessee at Chattanooga

Place of Publication

Chattanooga (Tenn.)

Abstract

The infinite potential well is one of the most well-known models in quantum mechanics, as well as the Dirac delta potential. In the first part of the thesis, one Dirac delta potential is considered using the time-independent Schr\"{o}dinger equation, the typical boundary conditions for an infinite well and normalization conditions to obtain the wave function $\psi(x)$. The system naturally imposes an additional condition at $\psi(0)$ due to the Dirac delta function, which yields the final equation for the energy levels $\{E_n\},$ that has infinitely many solutions. In the second part, a lattice potential within the infinite well is considered, which is defined as a finite sum of Dirac delta functions that are spread evenly within the bounds of the well. For future work, more numerical analysis should be done on these systems, as well as expanding the problem to several dimensions.

Acknowledgments

I am extremely grateful to Dr. Boris Belinskiy for his continued patience and feedback throughout this project as my primary advisor. I would also like to express my deepest gratitude to my defense committee, Dr. Christopher Cox, Dr. Lingju Kong, and Dr. Roger Nichols, for generously giving their time and knowledge. I am also grateful to my classmates and our administrative specialist Deborah Barr for all of their support and feedback through this entire project. Finally, I would like to thank my family, specifically my parents, my step-mom, and my brother, for always lending an ear and motivating me to see this thesis to the end.

Degree

M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science.

Date

5-2025

Subject

Differential equations; Dirac equation; Quantum theory; Schrödinger equation

Keyword

time-independent Schrodinger equation; infinite potential well; Dirac delta function; ordinary differential equations

Document Type

Masters theses

DCMI Type

Text

Extent

ix, 31 leaves

Language

English

Rights

http://rightsstatements.org/vocab/InC/1.0/

License

http://creativecommons.org/licenses/by/4.0/

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