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.