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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.06195 |
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| _version_ | 1866929609977626624 |
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| author | Zhou, Wanping Yang, Sanjiang Qiao, Haoxue |
| author_facet | Zhou, Wanping Yang, Sanjiang Qiao, Haoxue |
| contents | The relativistic corrections for the Dirac-Coulomb system are derived through the method of non-relativistic expansion. By expanding the large and small components of the Dirac wave function and the energy eigenvalues in terms of the square of the fine-structure constant $α^2$, we obtain iterative equations for calculating the higher-order relativistic corrections of Coulomb systems. For a single-electron system, the operator results of the iterative equations are consistent with those in the literature Ref[J.Phys.B,At.Mol.Opt.Phys.{\bf 56} 045001]. Using these iterative equations, we numerically calculate the relativistic corrections up to the order of $α^{20}$ for the hydrogen atom, which converge rapidly to the analytical results of the hydrogen atom. For the two-electron Dirac-Coulomb system, we also present iterative equations for calculating high-order energy corrections, as well as numerical energy corrections of ground state up to the order of $α^8$. This work also presents the non-relativistic expansion form of non-retarded Breit interaction correction. The $α^4$ order correction to the Dirac Coulomb energy and non-retarded Breit interaction corresponds precisely to the $α^4$ order relativistic correction. Higher-order expansion terms contribute at even powers of $α$, which represent the contributions from all Coulomb photons and single transverse photons under the non-retarded approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The non-relativistic expansion of Dirac-Coulomb energy and the non-retarded Breit interaction correction up to $α^8$order Zhou, Wanping Yang, Sanjiang Qiao, Haoxue Atomic Physics The relativistic corrections for the Dirac-Coulomb system are derived through the method of non-relativistic expansion. By expanding the large and small components of the Dirac wave function and the energy eigenvalues in terms of the square of the fine-structure constant $α^2$, we obtain iterative equations for calculating the higher-order relativistic corrections of Coulomb systems. For a single-electron system, the operator results of the iterative equations are consistent with those in the literature Ref[J.Phys.B,At.Mol.Opt.Phys.{\bf 56} 045001]. Using these iterative equations, we numerically calculate the relativistic corrections up to the order of $α^{20}$ for the hydrogen atom, which converge rapidly to the analytical results of the hydrogen atom. For the two-electron Dirac-Coulomb system, we also present iterative equations for calculating high-order energy corrections, as well as numerical energy corrections of ground state up to the order of $α^8$. This work also presents the non-relativistic expansion form of non-retarded Breit interaction correction. The $α^4$ order correction to the Dirac Coulomb energy and non-retarded Breit interaction corresponds precisely to the $α^4$ order relativistic correction. Higher-order expansion terms contribute at even powers of $α$, which represent the contributions from all Coulomb photons and single transverse photons under the non-retarded approximation. |
| title | The non-relativistic expansion of Dirac-Coulomb energy and the non-retarded Breit interaction correction up to $α^8$order |
| topic | Atomic Physics |
| url | https://arxiv.org/abs/2409.06195 |