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On The Quantization Problem In Curved Space, Benjamin Bernard

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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 …