Approximate Solutions to the Klein-Fock-Gordon Equation for the sum of Coulomb and Ring-Shaped like potentials

TL;DR

This paper provides approximate analytical solutions for the Klein-Fock-Gordon equation with Coulomb and ring-shaped potentials, revealing energy spectra, wave functions, and a dynamical symmetry group, bridging relativistic and nonrelativistic cases.

Contribution

It introduces an algebraic method using the $SU(1,1)$ symmetry to find energy spectra and wave functions for a relativistic particle in combined Coulomb and ring-shaped potentials.

Findings

Discrete and continuous energy spectra identified.

Analytical wave functions derived.

Relativistic results reduce to nonrelativistic limits as c→∞.

Abstract

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass$M$, described by the Klein-Fock-Gordon equation with equal scalar $S(\vec{r})$ and vector $V(\vec{r})$ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at $\left|E\right|<Mc^{2} $ and a continuous at $\left|E\right|>Mc^{2} $ energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group $SU(1,1)$ for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra and group generators in the limit $c\to \infty $ go over into the corresponding expressions for the nonrelativistic problem.