In the above, __ a__ = (0,0,a). This is a coulomb potential with the origin shifted to the imaginary point

This field has both charge and a magnetic moment. Dirac's equation describes the behaviour of particles with spin in an electromagnetic field. Hence the solution of the Dirac equation for this potential describes a quantum mechanical system in which there are two interacting particles with magnetic moments. There are no exact solutions of this type known, and the solution is of interest for this reason. It is also the case that the massless Kerr-Newman potential described here is the significant part of the interaction that an electron orbiting a micro black hole would feel. It is likely that micro black holes were created in the early universe, and may still be present today. A solution for the bound states of such a system predicts the spectral lines that would be expected from these black hole atoms, which may therefore be distinguishable from normal matter. The solution has applications to both of these problems.

The use of the Kerr-Newman potential enables a solution to the problem, since Chandrasekhar demonstrated that the Dirac equation separated for the full, massive Kerr-Newman potential. This is still true in the massless limit, but the coordinates in which it separates are oblate spheroidal coordinates rather than spherical polars.

As a sideline to the solution of the Dirac equation, I also considered the same problem for the Schrodinger and Klein-Gordon equations. The former is non-relativistic and neither equation accounts for the spin of the particle. For this reason it does not describe interacting objects with magnetic moments. However, the solution of these equations was a useful step towards the full Dirac solution and provides a comparison. The solution in both cases is computed numerically by using a combination of infinite continued fractions and numerical integration. In the Schrodinger case, it is possible to regard the potential as a perturbed coulomb potential. The perturbation is in fact infinite at the origin, and conventional perturbation theory does not work properly, which is interesting. A more careful treatment of the problem gave predicted perturbations in excellent agreement with the numerical values.

The solution to the Dirac equation is still in progress. The equations have been reduced to a solvable form. The main problem in this case is the boundary condition to impose at the disc. In the Schrodinger and Klein-Gordon cases, there is a single wave function which describes the probability density of the electron. Continuity through the disc then provides a natural boundary condition and divides the solution space into even and odd modes. In the Dirac case the wavefunction is described by a four component spinor, and the boundary condition is more difficult. Continuity of the probability density through the disc will probably provide the best boundary condition. As a model of a black hole atom, it is not clear that this boundary condition is appropriate, since in that case the event horizon provides a natural innner boundary. But the solution may still be applicable, and it is certainly relevant to the first problem.

Pekeris considered the same problem with the motivation of a black hole atom solution, and indeed he computed the bound state eigenvalues. However, he imposed a boundary condition that the solution should be regular at the imaginary point __ r__ = i