The proofs in a quantum mechanical d'Alembert Principle

The d' Alembert principle states that if a particle is constrained to a manifold, no work can be done normal to the manifold, however, quantum mechanics forbids the restraint of a particle. The constraint is replaced by an infinite potential and the Schödinger equation can be separated to produce a potential field on the manifold which is a function of the manifold's curvature. This is done for a one-dimensional curve and then for a general manifold. New work is presented as the case of a particle on a circle and the case of a product manifold are investigated.