Symmetries in the many-body problems, a method to find its analytical solution, and Helium atom spectrum

In this work it is shown that there are symmetries beyond the Euclidean group \(E\left(3\right)\) in 3-body problem, and by extension in many-body problem, with inverse squared distance inter particle force. The symmetries in 3-body problem form a group: \(SO\left(4\times3,2\times3\right)/\left(C\le...

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Bibliographic Details
Published inarXiv.org
Main Author Ambhire, Siddhesh C
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.01.2024
Subjects
Energy levels
Energy spectra
Helium atoms
Mathematical analysis
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Summary:In this work it is shown that there are symmetries beyond the Euclidean group \(E\left(3\right)\) in 3-body problem, and by extension in many-body problem, with inverse squared distance inter particle force. The symmetries in 3-body problem form a group: \(SO\left(4\times3,2\times3\right)/\left(C\left(3\times2\right)\right)\), where \(C\left(n\right)\) is the planar translation group in n dimensions, which forms its Spectrum-Generating group. Some of these quantities commute with the Hamiltonian. The existence of these conserved quantities was verified by calculating energy spectrum of the Helium atom. This method can also be used to find symmetries in many-body problem, and to calculate energy levels, and wave-functions of more complicated systems, which include every possible atomic and molecular systems in chemistry.
ISSN:2331-8422