Quantum Particles Constrained on Cylindrical Surfaces with Non-constant Diameter

TL;DR

This paper develops a theoretical model for one-electron states on cylindrical surfaces with varying diameters, revealing how geometry and magnetic fields influence electronic structures and inducing effects like bound states and Aharonov-Bohm phenomena.

Contribution

It introduces a novel formulation for electrons on curved cylindrical surfaces with non-uniform diameters, incorporating magnetic fields and analyzing specific geometries such as catenoids and sinusoidal tubules.

Findings

Existence of geometry-induced bound states.

Band structures affected by surface shape.

Magnetic fields induce Aharonov-Bohm effects.

Abstract

We present a theoretical formulation of the one-electron problem constrained on the surface of a cylindrical tubule with varying diameter. Because of the cylindrical symmetry, we may reduce the problem to a one-dimensional equation for each angular momentum quantum number $m$ along the cylindrical axis. The geometrical properties of the surface determine the electronic structures through the geometry dependent term in the equation. Magnetic fields parallel to the axis can readily be incorporated. Our formulation is applied to simple examples such as the catenoid and the sinusoidal tubules. The existence of bound states as well as the band structures, which are induced geometrically, for these surfaces are shown. To show that the electronic structures can be altered significantly by applying a magnetic field, Aharonov-Bohm effects in these examples are demonstrated.