Constrained Quantum Systems as an Adiabatic Problem

Abstract

We derive the effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) in the asymptotic limit where the restoring forces tend to infinity. In contrast to earlier works we consider at the same time the effects of variations in the constraining potential and the effects of interior and exterior geometry which appear at different energy scales and thus provide, for the first time, a complete picture ranging over all interesting energy scales. We show that the leading order contribution to the effective Hamiltonian is the adiabatic potential given by an eigenvalue of the confining potential well-known in the context of adiabatic quantum wave guides. At next to leading order we see effects from the variation of the normal eigenfunctions in form of a Berry connection. We apply our results to quantum wave guides and provide an example for the occurrence of a topological phase due to the geometry of a quantum wave circuit, i.e. a closed quantum wave guide.

19 pages, 4 figures

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• Quantum gases on a torus
• The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces
• Spin-deformation coupling in two-dimensional polar materials
• All order covariant tubular expansion
• Extrinsic curvature effects in brane-world scenarios
• On a semi-classical limit of loop space quantum mechanics
• Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle