NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2007, Vol.10, No.4, pp.312-334


On Solutions of Schrödinger and Dirac Equations in Einstein Stationary Space-Time, Spherical, and Elliptical Models.
V. M. Red'kov

Exact solutions of the Schrödinger and Dirac equations in generalized cylindrical coordinates of the 3-dimensional space of positive constant curvature, spherical model, have been obtained. It is shown that all basis Schrödinger's and Dirac's wave functions are finite, single-valued, and continuous everywhere in spherical space model S3. The used coordinates are simply referred to Euler's angle variables , parameters on the unitary group SU(2), which permits to express the constructed wave solutions in terms of Wigner's functions . Specification of the analysis to the case of elliptic, SO(3.R) group space, model has been done. In so doing, the results substantially depend upon the spin of a particle. In scalar case, the part of the Schrödinger wave solutions must be excluded by continuity considerations, remaining functions are continuous everywhere in the elliptical 3-space. The latter is in agreement with the known statement: the Wigner functions at j = 0,1,2,... make up a correct basis in SO(3.R) group space. For the fermion case, it is shown that no Dirac solutions, continuous everywhere in elliptical space, do exist. Description of a Dirac particle in elliptical space of positive constant curvature cannot be correct in the sense of continuity adjusted with its topological structure.
Key words: spherical, elliptical, geometry, Shcrödinger and Dirac equations, Wigner functions, Euler angles, continuity, curvature, spin, topology

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