Scalar Curvature Factor in the Schroedinger Equation and Scattering on a Curved Surface

TL;DR

This paper investigates how the scalar curvature term in the quantum Hamiltonian influences particle scattering on curved surfaces, proposing that scattering data could resolve factor ordering ambiguities in quantum mechanics.

Contribution

It introduces the role of scalar curvature in the Schrödinger equation on curved surfaces and explores its potential to experimentally determine factor ordering ambiguities.

Findings

Scalar curvature affects scattering cross sections.

Magnetic fields influence scattering sensitivity.

Potential to resolve theoretical ambiguities experimentally.

Abstract

The scattering of free particles constrained to move on a cylindrically symmetric curved surface is studied. The nontrivial geometry of the space contributes to the scattering cross section through the kinetic as well as a possible scalar curvature term in the quantum Hamiltonian. The coefficient of the latter term is known to be related to the factor ordering problem in curved space quantization. Hence, in principle, the scattering data may be used to provide an experimental resolution of the theoretical factor ordering ambiguity. To demonstrate the sensitivity required of such an experimental setup, the effect of a localized magnetic field in the scattering process is also analyzed.