A deep neural network approach to solve the Dirac equation

TL;DR

This paper introduces a deep neural network approach combined with the inverse Hamiltonian method to accurately solve the Dirac equation for both ground and excited states, overcoming limitations of traditional variational methods.

Contribution

It extends previous neural network methods to include excited states and compares two different approaches, enhancing the computational tools for relativistic quantum systems.

Findings

Successfully computed low-lying states for Coulomb potential

Validated method with Woods-Saxon potential

Demonstrated advantages of proposed approaches

Abstract

We extend the method from [Naito, Naito, and Hashimoto, Phys. Rev. Research 5, 033189 (2023)] to solve the Dirac equation not only for the ground state but also for low-lying excited states using a deep neural network and the unsupervised machine learning technique. The variational method fails because of the Dirac sea, which is avoided by introducing the inverse Hamiltonian method. For low-lying excited states, two methods are proposed, which have different performances and advantages. The validity of this method is verified by the calculations with the Coulomb and Woods-Saxon potentials.