Publication Date


Document Type


Committee Members

Gary Farlow (Committee Member), Lok Lew Yan Voon (Advisor), Doug Petkie (Other), Morten Willatzen (Committee Member)

Degree Name

Master of Science (MS)


The nonrelativistic quantum mechanics of particles constrained to curved surfaces is studied. There is open debate as to which of several approaches is the correct one. After a review of existing literature and the required mathematics, three approaches are studied and applied to a sphere, spheroid, and triaxial ellipsoid.

The first approach uses differential geometry to reduce the problem from a three dimensional problem to a two-dimensional problem. The second approach uses three dimensions and holds one of the separated wavefunctions and its associated coordinate constant. A third approach constrains the particle in a three-dimensional space between two parallel surfaces and takes the limit as the distance between the surfaces goes to zero.

Analytic methods, finite element methods, and perturbation theory are applied to the approaches to determine which are in agreement. It is found that the differential geometric approach has the most agreement.

Constrained quantum mechanics has application in materials science, where topological surface states are studied. It also has application as a simplified model of Carbon-60, graphene, and silicene structures. It also has application as in semiclassical quantum gravity, where spacetime is a pseudo-Riemannian manifold, to which the particles are constrained.

Page Count


Department or Program

Department of Physics

Year Degree Awarded


Included in

Physics Commons