Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems

Abstract

We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the external curvature of the constraint manifold, the (Riemannian) curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is an example of geometric phase, closely related to that originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the external curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.

27 pages with 1 figure, submitted to Phys. Rev. A

Cited by in corpus (32)

• Little Magnetic Book
• Effective Hamiltonian model for helically constrained quantum systems within adiabatic perturbation theory: application to the Chirality-Induced Spin Selectivity (CISS) Effect
• Quantum graphs as holonomic constraints
• Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential
• How does geometry affect quantum gases?
• Quantum strips on surfaces
• PT-symmetric models in curved manifolds
• Curvature-induced quantum behaviour on a helical nanotube
• Effective Hamiltonians for Constrained Quantum Systems
• Constrained Quantum Systems as an Adiabatic Problem
• Effective Hamiltonians for Thin Dirichlet Tubes with Varying Cross-Section
• Integrated density of states and Wegner estimates for random Schrödinger Operators
• Semiclassical analysis of constrained quantum systems
• Induced magnetic moment for a spinless charged particle in the thin-layer approach
• Generalised Quantum Waveguides
• Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides
• Canonical quantization of motion on submanifolds
• Torsion-induced persistent current in a twisted quantum ring
• Quantum dynamics of a particle constrained to lie on a surface
• Anomalous phase shift in a twisted quantum loop
• The adiabatic limit of Schrödinger operators on fibre bundles
• Dirac Equation on a Curved 2+1 Dimensional Hypersurface
• Quantization Ambiguity, Ergodicity, and Semiclassics
• Flux-free conductance modulation in a helical Aharonov-Bohm interferometer
• Quantum gases on a torus
• All order covariant tubular expansion
• Wires with Quantum Memory
• Brownian Motion Model of Quantization Ambiguity and Universality in Chaotic Systems
• Curvature induced quantum potential on deformed surfaces
• Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle
• On a semi-classical limit of loop space quantum mechanics
• Quantum Mechanics on Curved Hypersurfaces