Sturm-Liouville theory with applications to quantum mechanics

This expository thesis presents the primary theorems that together form what is known as Sturm-Liouville theory for a boundary value, linear second order differential equation involving a weight function. The key theorems regarding the self-adjoint condition of the Sturm-Liouville equation are developed in a manner that builds the theory. The Dirac delta function is used in a procedure for finding the Green's function for the Sturm-Liouville equation. A recursive process is developed that uses the Green's function to construct an L2 -complete set of eigenfunctions. The connection between the self-adjoint Sturm-Liouville equation and the Hermitian operators of mathematical physics is explored in physical applications of the theory. The infinitely deep square well from quantum mechanics is presented as an example of a regular Sturm-Liouville problem. Various methods used in solving singular Sturm-Liouville cases are illustrated by examples such as the quantum harmonic oscillator and the hydrogen atom.