Thin Layer Quantization Method for a Spin Particle on a Curved Surface

Using the fundamental framework of the thin-layer quantization method, we discuss the non-relativistic limit of the Schrödinger-Dirac equation for a particle constrained to move on a curved surface. We show that the inclusion of spin connections in the formalism give rise to scalar terms which provide a new scalar geometric potential. The coupling between the spin connections determined by the geometry of the curved surface and the spin of the particle can generate bound states even for the repulsive case of this obtained geometric potential. The developed procedure is applied to a surface of axial symmetry. We give three interesting examples of surface confinement, namely cylindrical, spherical and conical, and we explicitly deduce the energy levels for each case.