The Dirac equation with a Coulomb potential in D dimensions

The Dirac equation is generalized to D + 1 spacetime case. The radial equations of this quantum system are obtained and solved exactly by the Tricomi equation approach. The energy levels E(n, l, D) are analytically presented. The dependences of the energy E(n, l, D) on the dimension D are also analysed. 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. Unexpectedly, there is an absence of bound states for this quantum system with the special case D = 1, which is explained in some detail from the fact that the eigenvalues and eigenfunctions do not exist for D = 1.