On solutions of Schrodinger and Dirac equations in spaces of constant curvature, spherical and elliptical models

TL;DR

This paper derives exact solutions for Schrödinger and Dirac equations in spherical and elliptical spaces of constant curvature, revealing how particle spin influences the continuity and form of wave functions in these geometries.

Contribution

It provides explicit wave function solutions in spherical and elliptical models using Euler angles and Wigner functions, highlighting topological effects on fermion solutions.

Findings

Wave functions are finite, single-valued, and continuous in spherical space.

In elliptical space, scalar solutions are continuous, but Dirac solutions are not.

Fermion solutions cannot be globally continuous in elliptical space due to topological constraints.

Abstract

Exact solutions of the Schrodinger 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 Schrodinger's and Dirac's wave functions are finite, single-valued, and continuous everywhere in spherical space model S_{3}. The used coordinates (\rho, \phi,z) are simply referred to Eiler's angle variables (\alpha, \beta, \gamma), parameters on the unitary group SU(2), which permits to express the constructed wave solutions \Psi(\rho, phi,z) in terms of Wigner's functions $D_{mm'}^{j}(\alpha, \beta, \gamma)$. 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 the particle. In scalar case, the part of the Schrodinger wave solutions must be excluded by continuity considerations, remaining functions are continuous everywhere in the elliptical 3-space. The latter is in agrement with the known statement: the Wigner functions D_{mm'}^{j}(\alpha, \beta, \gamma) 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 the Dirac particle in elliptical space of positive constant curvature cannot be correctly in the sense of continuity adjusted with its topological structure.