Non-relativistic Quantum Mechanics on a Twisted Cylindrical Surface

TL;DR

This work analyzes a non-relativistic electron confined to a twisted cylindrical surface using the da Costa geometric-potential framework derived from the Green-Lagrange strain. It shows that linear and certain non-linear torsions induce a geometric phase in the wavefunction, while leaving the bound-state energy spectrum and surface probability density unchanged, and that scattering transmission is largely independent of torsion but sensitive to angular momentum and radius. The approach combines elasticity theory with quantum confinement on curved manifolds to reveal a separation between phase effects and energy shifts, with implications for curvature-engineered nanostructures. The results offer a pathway to control quantum states via geometric design in curved nanostructures while maintaining robust energy spectra, potentially impacting device engineering in curved quantum systems.

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 Schrödinger 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.

Figures (6)

• Figure 1: Schematic of the twisted cylinder model used in this work. The diagram shows the helical deformation of the coordinate lines and the key parameters.
• Figure 2: Analysis of the effective potential ($V_{\text{eff}}$), assuming the free electron mass ($m^*\approx m_e$). (a, b)$V_{\text{eff}}$ as a function of the longitudinal coordinate $z$ for different values of $l$ and $\alpha$.
• Figure 3: Illustration of the scattering problem. (a) Schematic of the physical setup, where a finite twisted cylinder section (Region II) of length $L$ is embedded in an infinite untwisted cylinder (Regions I and III). (b) The corresponding effective potential profile $V_{\text{eff}}$ encountered by a particle with a given angular momentum mode. The torsion creates a potential step which acts as a scattering center.
• Figure 4: Transmission probability ($T$) versus incident energy ($E$). (a) The transmission is independent of the torsion parameter $\alpha$. (b) The energy threshold for transmission increases for higher angular momentum modes $l$. (c) The energy threshold decreases for a larger cylinder radius $R$. Fixed parameters are indicated in the plots.
• Figure 5: Illustration of the scattering problem. Schematic of the physical setup, where a finite twisted cylinder section (Region II) of length $L$ is embedded in an infinite free space (Regions I and III).