Relativistic Wave Equations: An Operational Approach

TL;DR

This paper demonstrates how algebraic operator methods can effectively solve various relativistic wave equations, providing exact or approximate solutions and enabling practical numerical algorithms for particles under different potentials.

Contribution

It introduces an operational algebraic approach to relativistic wave equations, offering a unified framework for exact, approximate, and numerical solutions.

Findings

Effective algebraic methods for relativistic Schrödinger, Klein-Gordon, and Dirac equations.

Development of numerical algorithms for particles in non-trivial potentials.

Demonstration of solutions' accuracy and computational efficiency.

Abstract

The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schr\"odinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.