Realizing Holonomic Constraints in Classical and Quantum Mechanics

We consider the problem of constraining a particle to a smooth compact submanifold Σ of configuration space using a sequence of increasing potentials. We compare the classical and quantum versions of this procedure. This leads to new results in both cases: an unbounded energy theorem in the classical case, and a quantum averaging theorem. Our two step approach, consisting of an expansion in a dilation parameter, followed by averaging in normal directions, emphasizes the role of the normal bundle of Σ, and shows when the limiting phase space will be larger (or different) than expected.