Gauge fields and extrapotentials in constrained quantum systems - art. no.042112

We derive an effective Hamiltonian for a quantum system constrained to a su bmanifold (the constraint manifold) of configuration space (the ambient spa ce) by an infinite restoring force. We pay special attention to how this Ha miltonian depends on quantities which are external to the constraint manifo ld, such as the extrinsic curvature of the constraint manifold, the curvatu re of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential ap pears as a gauge potential in the constrained Hamiltonian. This gauge poten tial is closely related to the geometric phase originally discussed by Berr y. The constrained Hamiltonian also contains an effective potential dependi ng an the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The gene ral nature of our analysis allows applications to a wide variety of problem s, such as rigid molecules, the evolution of molecular systems along reacti on paths, and quantum strip waveguides.