Solution of One-dimensional Dirac Equation via Poincare Map

TL;DR

This paper introduces a Poincare Map method for solving the one-dimensional Dirac equation that is highly accurate, efficient, and easy to implement numerically, validated by comparison with exact solutions.

Contribution

The paper presents a novel recursive Poincare Map approach for solving the 1D Dirac equation without approximating spatial derivatives, enhancing numerical efficiency.

Findings

Method shows rapid convergence and high accuracy.

Comparison with analytical solutions confirms validity.

Approach is practical for numerical implementation.

Abstract

We solve the general one-dimensional Dirac equation using a "Poincare Map" approach which avoids any approximation to the spacial derivatives and reduces the problem to a simple recursive relation which is very practical from the numerical implementation point of view. To test the efficiency and rapid convergence of this approach we apply it to a vector coupling Woods--Saxon potential, which is exactly solvable. Comparison with available analytical results is impressive and hence validates the accuracy and efficiency of this method.