Abstract:
This research provides new insights into the thin-layer quantization method for both spinless and spin-1/2 particles, particularly regarding the impact of the induced geometric potential on a system's energy levels. For the spinless case, we have correctly applied the thin-layer quantization scheme to a particle constrained to a cone. The bound states for this system occur only when the quantum number is equal to zero, and we demonstrate that the Gaussian curvature reduces the binding energy.
For the spin-1/2 particle case, we have considered the non-relativistic limit of the Schrödinger-Dirac equation defined in a curved space-time. This was achieved by introducing a squeezing potential through a minimal coupling procedure. As a result, we obtained a new scalar geometric potential composed of three additive terms. These new potentials can alter the nature of the usual da Costa potential. Furthermore, we have demonstrated, through examples, the important role of the coupling between the geometry of the curved surface and the particle's spin in creating bound states.
