Quantum Scattering of Spinless Particles in Riemannian Manifolds

Abstract

Quantum mechanics is sensitive to the geometry of the underlying space. Here, we present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space. When the motion manifold hosts localized curvature modulations, scattering occurs from an emergent geometric potential and the metric tensor field. Analytical and full numerical simulations identify the geometric potential as the primary source for low-energy scattering, while the metric tensor field of the curved space governs high-energy diffraction. Compared to flat spaces, important differences in the validity range of perturbation approaches are found and demonstrated by full numerical simulations using combined finite element and boundary element methods. As an illustration, we consider a Gaussian-shaped dent leading to effects known as gravitational lensing. Experimentally, the considered setup is realizable based on geometrically engineered 2D materials.

15 pages, 5 figures

References in corpus (6)

• Quantum mechanics on curved 2D systems with electric and magnetic fields
• Observation of ultracold atomic bubbles in orbital microgravity
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• Magnetic and geometric effects on the electronic transport of metallic nanotubes
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Cited by in corpus (2)

• Quantum particle on the surface of a spherocylindrical capsule
• Geometrodynamics of a 2D Curved Surface due to a Constrained Quantum Particle via its Gravitational Dual: ${\mathcal{S}}^2$ Analytical Model Calculations