Quantum Mechanics on Curved Hypersurfaces

TL;DR

This paper investigates quantum mechanics equations on curved hypersurfaces using two different methods, comparing their results to understand the effects of geometry on quantum particles.

Contribution

It introduces and compares the thin layer method and Dirac's quantization procedure for quantum equations on curved hypersurfaces, highlighting their differences.

Findings

Comparison of the two methods' results.

Insights into geometric effects on quantum particles.

Notes on differences between the methods.

Abstract

In this work, Schr\"odinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac's quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared and some notes on the differences between the results will be included.