Scattering due to geometry: The case of a spinless particle moving on an asymptotically flat embedded surface

Abstract

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.

arXiv comment12 pages, 5 figures, to appear in Phys. Rev. A

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• The geometric potential of a double-frequency corrugated surface
• Scattering by a collection of $\delta$-function point and parallel line defects in two dimensions
• Sphere on a plane: Two-dimensional scattering from a finite curved region
• Geometric scattering in the presence of line defects
• Quantum particle on a surface: Catenary surface and Paraboloid of revolution