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

TL;DR

This paper investigates how particles move and scatter on a surface composed of a plane with a spherical bump, revealing classical and quantum behaviors including interference effects akin to a double slit.

Contribution

It provides exact solutions for classical, semi-classical, and quantum scattering on a curved surface with uniform curvature, highlighting the interplay between geometry and quantum interference.

Findings

Classical motion follows geodesics independent of velocity.

Quantum scattering depends on particle energy.

Semi-classical analysis explains quantum interference patterns.

Abstract

Non-relativistic 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, semi-classical, and fully quantum problems, which are all exactly solvable, and show how semi-classical 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.