Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}

Details Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3} Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}

2011-08-31

arxiv.org/abs/1108.6176v1

J-NPCS Vol. 14, no 2. (2011) pp 1-20

Physics

Quantum Physics

28 pages, 2 figures, 60 references

Scientific paper

The Coulomb problem for Schr\"{o}dinger equation is examined, in spaces of constant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base of generalized parabolic coordinates. In contrast to the hyperbolic case, in spherical space S_{3} such parabolic coordinates turn to be complex-valued, with additional constraint on them. The technique of the use of such real and complex coordinates in two space models within the method of separation of variables in Schr\"{o}dinger equation with Kepler potential is developed in detail; the energy spectra and corresponding wave functions for bound states have been constructed in explicit form for both spaces; connections with Runge-Lenz operators in both curved space models are described.

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