The Klein–Gordon Equation with a Coulomb Potential in D Dimensions

Abstract

The solutions of the Klein–Gordon equation with a Coulomb potential in D dimensions are obtained exactly, and the energy levels E(n,l,D) are analytically presented. The dependence of the energy difference ΔE(n,l,D) for D and D-1 on the dimension D is demonstrated as three different kinds of rules. The dependence of the energy E(n,l,D) on the dimension D is also analyzed. It is shown that the energy E(n,l,D) (l≠0) is almost independent of the quantum number l, while E(n,0,D) first decreases and then increases as the dimension D increases. There is no bound state of the S-wave for D=2.