Quantum mechanics of fermion confined to a curved surface in Foldy-Wouthuysen representation

In Foldy-Wouthuysen representation, we deduce the effective quantum mechanics for a particle confined to a curved surface by using the thin-layer quantization scheme. We find that the spin effect caused by confined potential as the results of relativistic correction in the non-relativistic limit. Furthermore, the spin connection appeared in curved surface which depends on curvature contributes a Zeeman-like gap in the relativistic correction term. In addition, the confined potential also induces a curvature-independent energy shift, which is from the zitterbewegung effect. As an example, we apply the effective Hamiltonian to torus surface, in which we obtain expectantly the spin effects related to confined potential. Those results directly demonstrate the scaling of the uncommutation of the non-relativistic limit and the thin-layer quantization formalism