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Abstract <p>This article presents a novel method for solving a relativistic wave equation for a spin-<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1/2\)</EquationSource> <!--BPhysMGU2670007Kalitenko-m3--> </InlineEquation> massive particle. Relativistic wavefunctions are obtained using neural networks, and a novel approach to solving the Klein–Gordon–Fock equation has been developed, inspired by the ancient method of Heron. Unlike the four-component Dirac bispinor, this approach uses two-component wavefunctions. The method has been validated through application to two well-known physical systems: a hydrogen-like atom and a charged particle in a uniform magnetic field. The obtained energy spectra are in complete agreement with the analytical solutions derived from the Dirac equation. The method can be applied to more complex problems in quantum physics and chemistry.</p>

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Heron-Inspired Method to Solve a Relativistic Wave Equation for a Spin-\(\boldsymbol{1/2}\) Massive Particle

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • A. M. Kalitenko,
  • P. I. Pronin

摘要

Abstract

This article presents a novel method for solving a relativistic wave equation for a spin- \(1/2\) massive particle. Relativistic wavefunctions are obtained using neural networks, and a novel approach to solving the Klein–Gordon–Fock equation has been developed, inspired by the ancient method of Heron. Unlike the four-component Dirac bispinor, this approach uses two-component wavefunctions. The method has been validated through application to two well-known physical systems: a hydrogen-like atom and a charged particle in a uniform magnetic field. The obtained energy spectra are in complete agreement with the analytical solutions derived from the Dirac equation. The method can be applied to more complex problems in quantum physics and chemistry.