Symmetries of Relativistic Hydrogen Atom
DOI:
https://doi.org/10.15407/ujpe64.12.1148Keywords:
Dirac equation, Coulomb interaction, hydrogen atom, relativistic quantum mechanics, symmetryAbstract
The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.
References
V. Fock. Zur Theorie des Wasserstoffatoms. Z. Phys. 98, 145 (1935). https://doi.org/10.1007/BF01336904
V. Bargmann. Zur Theorie des Wasserstoffatoms. Bemerkungen zur gleichnamigen Arbeit von V. Fock. Z. Phys. 99, 576 (1936). https://doi.org/10.1007/BF01338811
P.A.M. Dirac. The quantum theory of the electron. Proc. Roy. Soc. Lond. A. 117, 610 (1928). https://doi.org/10.1098/rspa.1928.0023
M.H. Johnson, B.A. Lippmann. Relativistic Kepler problem. Phys. Rev. 78, 329 (1950).
E. De Groot. The virial theorem and the Dirac H atom. Am. J. Phys. 50, 1141 (1982). https://doi.org/10.1119/1.12917
A.A. Stahlhofen. Algebraic solutions of relativistic Coulomb problems. Helv. Phys. Acta 70, 372 (1997).
J-L. Chen, D-L. Deng, M-G. Hu. SO(4) symmetry in the relativistic hydrogen atom. Phys. Rev. A. 77, 034102 (2008). https://doi.org/10.1103/PhysRevA.77.034102
A.A. Stahlhofen. Comment on "SO(4) symmetry in the relativistic hydrogen atom". Phys. Rev. A. 78, 036101 (2008). https://doi.org/10.1103/PhysRevA.78.036101
W. Pauli. On the conservation of the lepton charge. Nuovo Cim. 6, 204 (1957). https://doi.org/10.1007/BF02827771
F. G¨ursey. Relation of charge independence and baryon conservation to Pauli's transformation. Nuov. Cim. 7, 411 (1958). https://doi.org/10.1007/BF02747705
I.Yu. Krivsky, V.M. Simulik. The Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations. Theor. Math. Phys. 90, 265 (1992). https://doi.org/10.1007/BF01036532
A.G. Nikitin. Superalgebras of symmetry operators for Coulomb and Aharonov-Bohm-Coulomb systems. In: Photon and Poincar'e group (Nova Sci., 1999) [ISBN:9781560727187].
Th.W. Ruijgrok. On the relativistic hydrogen atom. Acta Phys. Pol. 87 43 (1976).
V.M. Simulik, I.Yu. Krivsky. Clifford algebra in classical electrodynamical hydrogen atom model. Adv. Appl. Cliff. Algebras 7, 25 (1997). https://doi.org/10.1007/BF03041213
V.M. Simulik, I.Yu. Krivsky, I.L. Lamer. Bosonic symmetries, solutions and conservation laws for the Dirac equation with nonzero mass. Ukr. J. Phys. 58, 523 (2013). https://doi.org/10.15407/ujpe58.06.0523
V.M. Simulik. On the gamma matrix representations of SO(8) and Clifford algebras. Adv. Appl. Cliff. Algebras 28, 93 (2018). https://doi.org/10.1007/s00006-018-0906-3
V.M. Simulik, I.Yu. Krivsky. On the extended real Clifford-Dirac algebra and new physically meaningful symmetries of the Dirac equation with nonzero mass. Dopov. NAN Ukr. No. 5, 82 (2010) (in Ukrainian).
I.Yu. Krivsky, V.M. Simulik. Fermi-Bose duality of the Dirac equation and extended real Clifford-Dirac algebra. Cond. Matt. Phys. 13, 43101 (2010). https://doi.org/10.5488/CMP.13.43101
V.M. Simulik, I.Yu. Krivsky, I.L. Lamer. Application of the generalized Clifford-Dirac algebra to the proof of the Dirac equation Fermi-Bose duality. TWMS J. App. Eng. Math. 3, 46 (2013). https://doi.org/10.1109/MMET.2012.6331206
V.M. Simulik, I.Yu. Krivsky. Bosonic symmetries of the Dirac equation. Phys. Lett. A. 375, 2479 (2011). https://doi.org/10.1016/j.physleta.2011.03.058
B. Wybourne. Classical Groups for Physicists (Wiley, 1974) [ISBN: 978-0471965053].
J. Elliott, P. Dawber. Symmetry in Physics (Macmillian Press, 1979), Vol. 1 [ISBN: 978-0333382707]. https://doi.org/10.1007/978-1-349-07635-2_1
L.L. Foldy, S.A. Wouthuysen. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78, 29 (1950). https://doi.org/10.1103/PhysRev.78.29
V.M. Simulik, I.O. Gordiyevich. On the symmetry of relativistic hydrogen atom and the Foldy-Wouthuysen representation. In: Abstracts of the Reports of the Intern. Conference of Young Scientists and Post-Graduates (Institute of Electron Physics, 2013).
Downloads
Published
How to Cite
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
