Math Physics VIGRE group

Cal Burgoyne and Robert Varley

Mathematical techniques in solving quantum mechanical problems

Abstract: We will study the eigenvalue problems that arise from the Schroedinger equation in quantum mechanics. The physical problem is usually specified by a particular potential energy function and that leads to the mathematical problem of finding the eigenfunctions for the energy operator. We will introduce the quantum mechanical and mathematical ideas as we go. We can start with some interesting 1-dimensional potentials, such as the barrier potential showing tunneling effects in quantum mechanics, and the quadratic potential for a quantum harmonic oscillator. Then as time permits we can lead up to more complicated potentials, such as the Coulomb potential of a nucleus which governs the electron orbitals and leads to the structure of the periodic table for the elements.

Prerequisites: several variable calculus and linear algebra.

Workload: Intermediate between attending a weekly seminar and taking a 3 credit hour class with no tests. The participants will prepare and present material as the group progresses through some planned topics. An average of 1-2 hours per week outside the seminar should suffice to keep up; and the demands of preparing the presentations can be shared and distributed, so that they might average out to another 1-2 hours.