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Moscow University Physics Bulletin

A Neural Network Approach to Solve the Relativistic Wave Equation for Spinless Particles

A. M. Kalitenko

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract In this work, we discuss new ways to incorporate physical information into neural network training and describe modern neural network methods for quantum systems and their extensions. We describe an algorithm for solving the stationary Klein–Gordon–Fock equation using the neural network quantum states (NNQS) method. The method has been tested on two well-known systems: a relativistic spinless particle in a Coulomb potential and a charged particle in a uniform magnetic field.

DOI
10.3103/s002713492570273x

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Abstract In this work, we discuss new ways to incorporate physical information into neural network training and describe modern neural network methods for quantum systems and their extensions. We describe an algorithm for solving the stationary Klein–Gordon–Fock equation using the neural network quantum states (NNQS) method. The method has been tested on two well-known systems: a relativistic spinless particle in a Coulomb potential and a charged particle in a uniform magnetic field.
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Heron-Inspired Method to Solve a Relativistic Wave Equation for a Spin-1/2\boldsymbol{1/2}1/2 Massive Particle
This article presents a novel method for solving a relativistic wave equation for a spin- 1/21/21/2 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.
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