Geometric effects resulting from square and circular confinements for a particle constrained to a space curve

TL;DR

This paper explores how different confinement geometries, square and circular, influence the quantum behavior of a particle on a space curve, revealing unique geometric potentials, gauge fields, and spin interactions.

Contribution

It introduces novel geometric potentials and gauge fields arising from specific confinements and analyzes their effects on particle dynamics, including spin and electromagnetic interactions.

Findings

Square confinement induces torsion and curvature-related geometric potentials.

Circular confinement results in a geometric gauge potential and magnetic moment coupling.

Geometric effects significantly influence particle behavior on constrained space curves.

Abstract

Investigating the geometric effects resulting from the detailed behaviors of the confining potential, we consider square and circular confinements to constrain a particle to a space curve. We find a torsion-induced geometric potential and a curvature-induced geometric momentum just in the square case, while a geometric gauge potential solely in the circular case. In the presence of electromagnetic field, a geometrically induced magnetic moment couples with magnetic field as an induced Zeeman coupling only for the circular confinement, also. As spin-orbit interaction is considered, we find some additional terms for the spin-orbit coupling, which are induced not only by torsion, but also curvature. Moreover, in the circular case, the spin also couples with an intrinsic angular momentum, which describes the azimuthal motions mapped on the space curve. As an important conclusion for the thin-layer quantization approach, some substantial geometric effects result from the confinement boundaries. Finally, these results are proved on a helical wire.