Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-12-17
Physics
High Energy Physics
High Energy Physics - Theory
10 pages, Proc. of 5th International Conference Bolyai-Gauss-Lobachevsky: Non-Euclidean Geometry In Modern Physics (BGL-5).
Scientific paper
By analogy with the Lobachevsky space H_{3}, generalized parabolic coordinates (t_{1},t_{2},\phi) are introduced in Riemannian space model of positive constant curvature S_{3}. In this case parabolic coordinates turn out to be complex valued and obey additional restrictions involving the complex conjugation. In that complex coordinate system, the quantum-mechanical Coulomb problem is stu- died: separation of variables is carried out and the wave solutions in terms of hypergeometric functions are obtained. At separating the variables, two parameters k_{1} and k_{2} are introduced, and an operator B with the eigen values (k_{1}+k_{2}) is found, which is related to third component of the known Runge-Lenz vector in space S_{3} as follows: i B = A _{3} + i \vec{L}^{2}, whereas in the Lobachevsky space as B =A_{3} + \vec{L}^{2}. General aspects of the possibility to employ complex coordinate systems in the real space model S_{3} are discussed.
Bogush Andrei A.
Otchik V. S.
Red'kov Victor M.
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