Sinc-based method for an efficient solution in the direct space of quantum wave equations with periodic boundary conditions

TL;DR

This paper introduces a sinc-based method for solving quantum wave equations in direct space, providing exact derivative representation and comparable accuracy to reciprocal space methods, demonstrated on graphene nanoribbons.

Contribution

The paper presents a novel sinc-based approach that achieves exact derivatives in direct space, improving efficiency and accuracy over traditional finite-difference methods.

Findings

Exact derivative representation in direct space using sinc functions

Equivalent accuracy to reciprocal space methods

Successful application to Dirac equation in graphene nanoribbons

Abstract

The solution of differential problems, and in particular of quantum wave equations, can in general be performed both in the direct and in the reciprocal space. However, to achieve the same accuracy, direct-space finite-difference approaches usually involve handling larger algebraic problems with respect to the approaches based on the Fourier transform in reciprocal space. This is the result of the errors that direct-space discretization formulas introduce into the treatment of derivatives. Here, we propose an approach, relying on a set of sinc-based functions, that allows us to achieve an exact representation of the derivatives in the direct space and that is equivalent to the solution in the reciprocal space. We apply this method to the numerical solution of the Dirac equation in an armchair graphene nanoribbon with a potential varying only in the transverse direction.