2011, Vol.14, No.2, pp.106-125
The Coulomb problem for the Schrödinger equation is examined in
spaces of constant curvature,
Lobachevsky H3 and Riemann S3 models, on the base of generalized parabolic coordinates.
Opposite to the hyperbolic case, in spherical space S3 such
parabolic coordinates turn to be complex-valued, with additional constraint imposed 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 the
Schrödinger equation with Coulomb potential is developed in
detail. The energy spectra and corresponding wave functions for
bound states have been constructed in an explicit form for both
spaces; connections with Runge-Lenz operators in both curved
space models are described.
Key words: spaces of constant curvature, complex coordinates, Coulomb field, separation of variables, Schrödinger equation, bound states
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