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arXiv:2205.09603 · Physical Review A

Sphere on a plane: Two-dimensional scattering from a finite curved region

J. R. Anglin, Étienne Wamba

Published Oct 20, 2022• 1 citation
arXivHTMLPDF

Abstract

Nonrelativistic particles that are effectively confined to two dimensions can, in general, move on curved surfaces, allowing dynamical phenomena beyond what can be described with scalar potentials or even vector gauge fields. Here we consider a simple case of piecewise uniform curvature: a particle moves on a plane with a spherical extrusion. Depending on the latitude at which the sphere joins the plane, the extrusion can range from an infinitesimal bump to a nearly full sphere that just touches the plane. Free classical motion on this surface of piecewise uniform curvature follows geodesics that are independent of velocity, while quantum mechanical scattering depends on energy. We compare classical, semiclassical, and fully quantum problems, which are all exactly solvable, and show how semiclassical analysis explains the complex quantum differential cross section in terms of interference between two classical trajectories: the sphere on a plane acts as a kind of double slit.

DOI
10.1103/physreva.106.042216

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Abstract The advances in cold atom experiments have allowed construction of confining traps in the form of curved surfaces. This opens up the possibility of studying quantum gases in curved manifolds. On closed surfaces, many fundamental processes are affected by the local and global properties, i.e. the curvature and the topology of the surface. In this paper, we study the problem of potential scattering on a spherical surface and discuss its difference with that on a 2D plane. For bound states with angular momentum m , their energies ( E m ) on a sphere are related to those on a 2D plane ( − ∣ E m , o ∣ ) as E m = − ∣ E m , o ∣ + E R m 2 − 1 3 + O r o 2 R 2 , where E R = ℏ 2 / ( 2 MR 2 ) , and R is the radius of the sphere. Due to the finite extent of the manifold, the phase shifts on a sphere at energies E ∼ E R differ significantly from those on a 2D plane. As energy E approaches zero, the phase shift in the planar case approaches 0, whereas in the spherical case it reaches a constant that connects the microscopic length scale to the largest length scale R .
A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low-density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.
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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. \textcopyright{} 1996 The American Physical Society.
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