https://doi.org/10.1140/epja/s10050-025-01630-5
Regular Article - Theoretical Physics
A deep neural network approach to solve the Dirac equation
1
College of Physics, Jilin University, 130012, Changchun, China
2
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), 351-0198, Wako, Japan
3
Department of Physics, Graduate School of Science, The University of Tokyo, 113-0033, Tokyo, Japan
a
tnaito@ribf.riken.jp
b
jianli@jlu.edu.cn
c
haozhao.liang@phys.s.u-tokyo.ac.jp
Received:
10
March
2025
Accepted:
27
June
2025
Published online:
15
July
2025
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.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
