Curvature induced bound state (E < 0) eigenvalues and eigenfunctions for a particle constrained to move on the surface of a torus are calculated. A limit on the number of bound states a torus with minor radius a and major radius R can support is obtained. A condition for mapping constrained particle wave functions on the torus into free particle wave functions is established. Pacs number(s): 03.65Ge, 68.65.-k 1. Introduction.

No abstract

The Schrodinger equation for an electron near an azimuthally symmetric curved surface Σ in the presence of an arbitrary uniform magnetic field B is developed. A thin layer quantization procedure is implemented to bring the electron onto Σ, leading to the well known geometric potential V C ∝ h 2 − k and a second potential that couples A N , the component of A normal to Σ to mean surface curvature, as well as a term dependent on the normal derivative of A N evaluated on Σ. Numerical results in the form of ground state energies as a function of the applied field in several orientations are presented for a toroidal model.

A basis set expansion is performed to find the eigenvalues and wave functions for an electron on a toroidal surface T 2 subject to a constant magnetic field in an arbitrary direction. The evolution of several low-lying states as a function of field strength and field orientation is reported, and a procedure to extend the results to include two-body Coulomb matrix elements on T 2 is presented.

The curvature potential arising from confining a particle initially in three-dimensional space onto a curved surface is normally derived in the hard constraint q → 0 limit, with q the degree of freedom normal to the surface. In this work the hard constraint is relaxed, and eigenvalues and wave functions are numerically determined for a particle confined to a thin layer in the neighborhood of a toroidal surface. The hard constraint and finite layer (or soft constraint) quantities are comparable, but both differ markedly from those of the corresponding two dimensional system, indicating that the curvature potential continues to influence the dynamics when the particle is confined to a finite layer. This effect is potentially of consequence to the modelling of curved nanostructures.