The Klein–gordon Equation With a Coulomb Plus Scalar Potential in D Dimensions

The solutions of the Klein–Gordon equation with a Coulomb plus scalar potential in D dimensions are exactly obtained. The energy E(n,l,D) is analytically presented and the dependence of the energy E(n,l,D) on the dimension D is analyzed in some detail. The positive energy E(n,0,D) first decreases and then increases with increasing dimension D. The positive energy E(n,l D)(l≠0) increases with increasing dimension D. The dependences of the negative energies E(n,0,D) and E(n,l,D)(l≠0) on the dimension D are opposite to those of the corresponding positive energies E(n,0,D) and E(n,l,D)(l≠0). It is found that the energy E(n,0,D) is symmetric with respect to D=2 for D∈(0,4). It is also found that the energy E(n,l,D)(l≠0) is almost independent of the angular momentum quantum number l for large D and is completely independent of the angular momentum quantum number l if the Coulomb potential is equal to the scalar one. The energy E(n,l D) is almost overlapping for large D.