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Geometric effects on a quantum particle constrained to a helix, catenary, helicoid, or catenoid

Jonas Gomes, F. F. da Silva Filho, Fernando Moraes, Renato Teixeira

Published Feb 28, 2021• 0 citations
OpenAlex

Abstract

Since electrons in a ballistic regime perceive a carbon nanotube or a graphene layer structure as a continuous medium, we can use the study of the quantum dynamics of one electron constrained to a curve or surface to obtain a qualitative description of the conduction electrons' behavior. The confinement process of a quantum particle to a curve or surface leads us, in the so-called da Costa's approach, to a geometry-induced potential (GIP) in the effective Schr\odinger equation. With these considerations, this work aims to study in detail the consequences of constraining a quantum particle to a helix, catenary, helicoid, or catenoid, exploring the relations between these curves and surfaces using differential geometry. Initially, we use the variational method to estimate the energy of the particle in its ground state, and then, we obtain better approximations with the use of the confluent Heun function through numerical calculations. Thus, we conclude that a quantum particle constrained to an infinite helix has its angular momentum quantized due to the geometry of the curve, while in the cases of the catenary, helicoid, and catenoid the particle can be found either in a single bound state or in excited states which constitute a continuous energy band.

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Since electrons in a ballistic regime perceive a carbon nanotube or a graphene layer structure as a continuous medium, we can use the study of the quantum dynamics of one electron constrained to a curve or surface to obtain a qualitative description of the conduction electrons' behavior. The confinement process of a quantum particle to a curve or surface leads us, in the so-called da Costa's approach, to a geometry-induced potential (GIP) in the effective Schr\odinger equation. With these considerations, this work aims to study in detail the consequences of constraining a quantum particle to a helix, catenary, helicoid, or catenoid, exploring the relations between these curves and surfaces using differential geometry. Initially, we use the variational method to estimate the energy of the particle in its ground state, and then, we obtain better approximations with the use of the confluent Heun function through numerical calculations. Thus, we conclude that a quantum particle constrained to an infinite helix has its angular momentum quantized due to the geometry of the curve, while in the cases of the catenary, helicoid, and catenoid the particle can be found either in a single bound state or in excited states which constitute a continuous energy band.
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.
Abstract We revisit the Schrödinger equation of a quantum particle that is confined on a curved surface. Inspired by the seminal work of da Costa (1980 Phys. Rev. A 23 1982) we find the field equation in a more convenient notation. The contribution of the principal curvatures in the effective binding potential on the surface is emphasized. Furthermore, using the so-called Monge-Gauge we construct the approximate Schrödinger equation for a flat surface with small fluctuations. Finally, the resulting Schrödinger equation is solved for some specific surfaces. In particular, we give exact solutions for a particle confined on a Catenary surface and a paraboloid of revolution.
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.
In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schr?dinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schr?dinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.
Abstract Combining the deviation between thin layers’ adjacent surfaces with the confining potential method applied to the quantum curved systems, we derive the effective Schrödinger equation describing the particle constrained within a curved layer, accompanied by a general geometric potential V gq composed of a compression-corrected geometric potential V gq * and a novel potential V gq * * brought by the deviation. Applying this analysis to the cylindrical layer emerges two types of deviation-induced geometric potential, resulting from the the cases of slant deviation and tangent deviation, respectively, which strongly renormalizes the purely geometric potential and contribute to the energy spectrum based on a very substantial deepening of bound states they offer.
We consider an {\em elastic} helical nanoribbon that can take on various conformations, and study the effect of placing a quantum particle on its curved surface. Using a modified Canham-Helfrich model for the elastic energy, we write down the local elastic potential for the ribbon in terms of its bending rigidity, mean curvature MMM and Gaussian curvature KKK. The Schrödinger equation of a particle confined to a {\em rigid} curved surface is found using da Costa's formulation. It has a purely quantum geometric potential which depends on MMM and KKK. The Schrödinger equation of a particle on an {\em elastic } curved surface will therefore have a total potential comprising quantum and elastic potentials. We compute MMM and KKK for a helical ribbon and derive the total potential which depends on the conformation and is thus geometric in nature. Defining a dimensionless quantity RHR_HRH​, we study the behavior of the total geometric potential as RHR_HRH​ is varied. In the absence of an electron, the elastic potential is positive and has a single positive maximum for all conformations. Further, a binormal helical ribbon conformation has the lowest potential, while the normal ribbon has the highest, with those of the intermediate ribbons lying in between these. Intriguingly, when a quantum particle is placed on the elastic ribbon, above a certain critical value of RHR_HRH​, the presence of the quantum geometric potential {\em reverses} this order. But localized states for the particle are not supported. Only above a second critical value of RHR_HRH​, localized states appear for all conformations. The injection of an electron on {\it any} given conformation of the elastic ribbon will induce a conformational transition to the normal ribbon conformation.