scholarly journals Symmetries of Relativistic Hydrogen Atom

2019 ◽  
Vol 64 (12) ◽  
pp. 1148
Author(s):  
V. M. Simulik ◽  
I. O. Gordievich

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.

2005 ◽  
Vol 339 (3-5) ◽  
pp. 259-268 ◽  
Author(s):  
R.P. Martínez-y-Romero ◽  
H.N. Núñez-Yépez ◽  
A.L. Salas-Brito

Author(s):  
E. M. Оvsiyuk ◽  
A. D. Koral’kov

The known systems of the radial equations describing the hydrogen atom on the basis of the Dirac equation in the Lobachevsky–Riemann spaces of constant curvature are investigated. In the both geometrical models, the differential equations of second order with six regular singular points are found, and their exact solutions of Frobenius type are constructed. To produce the quantization rule for energy values we use the known condition which separates the transcendental Frobenius solutions. This provides us with the energy spectra that are physically interpretable and are similar to those for the Klein–Fock–Gordon particle in these space models. These spectra are similar to those that previously have appeared in studying the same systems of the equations with the use of the semi-classical approximation.


1993 ◽  
Vol 34 (11) ◽  
pp. 4964-4974 ◽  
Author(s):  
Jeffrey M. Cohen ◽  
Boris Kuharetz

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