Constrained Quantum Systems as an Adiabatic Problem

TL;DR

This paper derives a comprehensive effective Hamiltonian for constrained quantum systems, incorporating geometric and potential variations across energy scales, and demonstrates topological effects in quantum wave guides.

Contribution

It provides the first complete derivation of the effective Hamiltonian considering both potential variations and geometric effects at multiple energy scales.

Findings

Effective Hamiltonian includes adiabatic potential and Berry connection effects.

Geometric variations can induce topological phases in quantum wave guides.

Application to quantum wave circuits illustrates topological phenomena.

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.