Spinless Particle with Darwin–Cox Structure in an External Coulomb Field

Abstract

Generalized Klein–Fock–Gordon equation for a spinless particle with the Darwin–Cox structure, which takes into account distribution of the electric charge of a particle inside a finite spherical region is studied in presence of an external Coulomb field. There have been constructed exact Frobenius type solutions of the derived equations, convergence of the relevant power series with 8-term recurrent relations has been studied. As an analytical quantization rule is taken the so-called transcendency conditions. It provides us with a 4-th order algebraic equation with respect to energy values, which has four sets of roots. One set of roots, 0 < En,k < 1, depending on the angular momentum n = 0, 1, 2, . . . and the main quantum number n = 0, 1, 2, . . . may be interpreted as corresponding to some bound states of the particle in a Coulomb field. In the same manner, a generalized nonrelativistic Schr¨odinger equation for such a particle is studied, the final results are similar.