Non-relativistic Quantum Mechanics on a Twisted Cylindrical Surface

TL;DR

This paper investigates how twisting a cylindrical surface affects quantum states and scattering behavior of electrons, revealing that twist induces a geometric phase but does not alter the energy spectrum or transmission probabilities significantly.

Contribution

It introduces a detailed analysis of quantum behavior on twisted cylindrical surfaces, including bound states and scattering, highlighting the role of geometry and strain in quantum device design.

Findings

Twist induces a geometric phase in the wave function.

Energy spectrum of bound states is independent of twist.

Transmission probability is unaffected by torsion but depends on angular momentum and radius.

Abstract

Twisted cylindrical tubes are important model systems for nanostructures, heterostructures, and curved quantum devices. In this work, we investigate the quantum behavior of an electron confined to a twisted cylindrical surface. By first calculating the strain tensor to obtain the induced surface metric, we employ da Costa's formalism to derive the geometry-induced quantum potential. This potential modifies the Schr\"odinger equation even in the absence of external forces, allowing us to determine the bound states and energy eigenvalues. This was made in the linear and non-linear torsion regime. Furthermore, we analyze two distinct scattering problems: (i) scattering within an infinite cylinder containing a twisted section, and (ii) scattering of a free particle incident upon a finite twisted cylinder. Our goal is to understand how geometry and strain influence the properties of analogous untwisted systems. It turns out that both the linear and non-linear twists yield to a geometric phase into the wave function, while the da Costa potential is kept unchanged. Consequently, the system supports bound states whose energie spectrum is twist independent. For both scattering problems, we find that the transmission probability is insensitive to torsion, whereas it is significantly affected by the particle angular momentum and the cylinder's radius, exhibiting distinct oscillatory behavior. These findings suggest relevant implications for engineering quantum devices based on materials with controlled curvature and twist.