Quantizing Constrained Systems: New Perspectives

Abstract

We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order hbar^2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order hbar^2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics, Feynman path integrals, and semiclassically in terms of adiabatic actions.

11 pages, 2 figures

Cited by in corpus (31)

• Quadrupole collective states within the Bohr collective Hamiltonian
• Geometric potential and transport in photonic topological crystals
• Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems
• Curvature induced toroidal bound states
• Absence of anomalous interactions in the quantum theory of constrained charged particles in presence of electrical currents
• Curvature-induced geometric potential in strain-driven nanostructures
• Geometric Momentum: the Proper Momentum for a Free Particle on a Two-dimensional Sphere
• Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space
• Tuning the electrical resistivity of semiconductor thin films by nanoscale corrugation
• Low-dimensional quantum gases in curved geometries
• Effective theory for deformed nuclei
• Low-Temperature Resistivity Anomalies in Periodic Curved Surfaces
• Time in quantum mechanics: A fresh look at the continuity equation
• Curvature-induced quantum behaviour on a helical nanotube
• Supersolid phases of bosonic particles in a bubble trap
• Hamiltonization of Nonholonomic Systems and the Inverse Problem of the Calculus of Variations
• Semiclassical analysis of constrained quantum systems
• Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides
• Curvature-induced bound states and coherent electron transport on the surface of a truncated cone
• Wave functions in the neighborhood of a toroidal surface; hard vs. soft constraint
• Effective Field Theory of Emergent Symmetry Breaking in Deformed Atomic Nuclei
• Bose-Einstein condensate in an elliptical waveguide
• Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects
• Quantum Scattering of Spinless Particles in Riemannian Manifolds
• The centripetal force law and the equation of motion for a particle on a curved hypersurface
• Scattering due to geometry: The case of a spinless particle moving on an asymptotically flat embedded surface
• Quantization Ambiguity, Ergodicity, and Semiclassics
• Two-dimensional electron gas under the effect of constrained potential and magnetic field in curved space
• Expansion of a quantum gas in a shell trap
• Geometric scattering in the presence of line defects
• Curvature induced quantum potential on deformed surfaces