Narrow Results

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.

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.

The Dirac equation is investigated in D+1 dimensional space-time. The radial equations of this quantum system are obtained and solved exactly by the confluent hypergeometric equation approach. The energy levels E(n,l,D) are analytically presented. For the continuous dimension D as proposed by Nieto, the dependences of the energy difference ΔE(n,l,D) for D and D-1 on the dimension D are demonstrated as three different kinds of change rules. The dependences of the energy E(n,l,D) on the dimension D are also discussed. It is found that the energies E(n,l,D) (l≠0) are almost independent of the quantum number l for a large D, while E(n,0,D) first decreases and then increases with the increasing dimension D. The dependences of the energies E(n,l,ξ) on the potential strength ξ are also studied for the given dimension D=3. We find that the energies E(n,l,ξ) decrease with ξ≤l+1.