Quantum particle confined to a thin-layer volume: Non-uniform convergence toward the curved surface

We clearly refine the fundamental framework of the thin-layer quantization procedure, and further develop the procedure by taking the proper terms of degree one in q3 (q3 denotes the curvilinear coordinate variable perpendicular to curved surface) back into the surface quantum equation. The well-known geometric potential and kinetic term are modified by the surface thickness. Applying the developed formalism to a toroidal system obtains the modification for the kinetic term and the modified geometric potential including the influence of the surface thickness.