  We revisit the Schr\"{o}dinger equation of a quantum particle that is
confined on a curved surface. Inspired by the novel work of R. C. T. da Costa
[1] we find the field equation in a more convenient notation. The contribution
of the principal curvatures in the effective binding potential on the surface
is emphasized. Furthermore, using the so-called Monge-Gauge we construct the
approximate Schr\"{o}dinger equation for a flat surface with small
fluctuations. Finally, the resulting Schr\"{o}dinger equation is solved for
some specific surfaces. In particular, we give exact solutions for a particle
confined on a Catenary surface and a paraboloid of revolution.
