Bryenton, K. R. (2016). Darboux-Crum Transformations, Supersymmetric Quantum Mechanics, and the Eigenvalue Problem. Charlottetown, PE: University of Prince Edward Island.

The Darboux transformation and its generalization, Crum's method, are tools used to generate exactly solvable eigenvalue problems. The focus of this project was on generating new classes of exactly solvable quantum potentials for the Schrodinger equation. The Schrodinger equation is a fundamental equation in quantum mechanics which describes the behaviour of non-relativistic particles. Prior to generating new classes of potentials, an examination of the underlying mathematical theory beneath Show moreThe Darboux transformation and its generalization, Crum's method, are tools used to generate exactly solvable eigenvalue problems. The focus of this project was on generating new classes of exactly solvable quantum potentials for the Schrodinger equation. The Schrodinger equation is a fundamental equation in quantum mechanics which describes the behaviour of non-relativistic particles. Prior to generating new classes of potentials, an examination of the underlying mathematical theory beneath both the Darboux transformation and supersymmetric quantum mechanics is conducted, including a detailed proof of their equivalence. Following the analysis of methods, one of the most significant molecular potentials used in physics to describe the interaction between two atoms, the Hulthen Potential, is examined. Using Darboux-Crum techniques and supersymmetry, an extended solvable class of Hulthen potentials are constructed. Furthermore an analysis of the Hermite differential equation, which appears in solving the quantum harmonic oscillator problem, will be conducted. Finally, some new and interesting results will be shared on the Crum-generalization of the shifted non-linear quantum harmonic oscillator. Show less