Variational approach to the Coulomb problem on a cylinder

TL;DR

This paper uses a variational method to analyze the bound state energy of two charges on a cylindrical surface, revealing how curvature affects Coulomb binding energy and applying the model to carbon nanotubes.

Contribution

It introduces a variational approach to the Coulomb problem on a cylinder, accurately capturing known limits and exploring curvature effects on binding energy.

Findings

Binding energy varies monotonically with curvature C

E_B is nearly insensitive to small curvature C

E_B diverges logarithmically as R approaches zero

Abstract

We evaluate, by means of variational calculations, the bound state energy E_B of a pair of charges located on the surface of a cylinder, interacting via Coulomb potential - e^2 / r . The trial wave function involves three variational parameters. E_B is obtained as a function of the reduced curvature C = a_0 / R, where a_0 is the Bohr radius and R is the radius of the cylinder. We find that the energetics of binding exhibits a monotonic trend as a function of C ; the known 1D and 2D limits of E_B are reproduced accurately by our calculation. E_B is relatively insensitive to curvature for small C . Its value is ~ 1% higher at C = 1 than at C = 0. This weak dependence is confirmed by a perturbation theory calculation. The high curvature regime approximates the 1D Coulomb model; within our variational approach, E_B has a logarithmic divergence as R approaches zero. The proposed variational method is applied to the case of donors in single-wall carbon nanotubes (SWCNTs).