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The Schr\"odinger Hamiltonian of a spin zero particle as well as the PauliHamiltonian with spin-orbit coupling included of a spin one-half particle inelectromagnetic fields that are confined to a curved surface embedded in athree-dimensional space spanned by a general Orthogonal Curvilinear Coordinate(OCC) are constructed. A new approach, based on the physical argument that uponsqueezing the particle to the surface by a potential, then it is the physicalgauge-covariant kinematical momentum operator (velocity operator) transverse tothe surface that should be dropped from the Hamiltonian(s). In both cases,theresulting Hermitian gauge-invariant Hamiltonian on the surface is free from anyreference to the component of the vector potential transverse to the surface,and the approach is completely gauge-independent. In particular, for the PauliHamiltonian these results are obtained exactly without any further assumptionsor approximations. Explicit covariant plug-and-play formulae for theSchr\"odinger Hamiltonians on the surfaces of a cylinder, a sphere and a torusare derived.

Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates