[401 ] THE EFFECT OF A MAGNETIC POLE ON THE ENERGY LEVELS OF A HYDROGEN-LIKE ATOM BY C. J. ELIEZER AND S. K. ROY Received 11 September 1961 1. Introduction. The possible existence of a magnetic pole h as been under discussion for many years. Dirac worked out in detail an electro-dynamics in which a single magnetic pole is assumed to exist (1,2). He has shown that the existence of such a pole leads naturally to the quantization of charge, the magnetic pole strength g and the electric charge e being related by the equation eg = \hcp, (1) where p is an integer, h is Planck's constant divided by 2n, and c is the velocity of light. The motion of an electron in the field of a magnetic pole was also investigated, non- relativistically by Dirac (l), and relativistically by Harish-Chandra(3). It has been found that in both the cases the electron has no bound state in the field of a magnetic pole. Dirac has also posed the question whether an elementary particle can have both a charge and a pole. It is of some interest to consider the behaviour of an electron in the field of such a particle. As a particular case we may consider a hydrogen-like atom in which the nucleus has in addition to the electric charge Ze also a pole strength g. Two possibilities may be anticipated, one that the levels would be shifted by a finite amount owing to the presence of the pole and the other that the presence of the pole has such a dominating effect on the electron that no bound state would exist. We have considered this question, non-relativistically, and have found that the first possibility holds. We have also calculated the finite shift in the energy levels. 2. The wave-equation. The electric and magnetic fields at a point (r, 6, <j>) due to a particle having a magnetic pole of strength g and an electric charge Ze at the origin, are given by ^ r = £e/r 2 , R, = g/r 2 , (2) = Ef = 0, H e = H^ = O.J These fields are obtainable from a scalar potential A o and a vector potential A having components A r , A e , A^, given by A 0 = Zer~ 1 , A r = 0, A g = 0, A^ = gr~ x tan \6. (3) We notice that A^ becomes infinite along the line 6 = n, which is the nodal line(l). The solutions are valid at all other points. The wave equation for an electron in the field (A o , A) corresponding to a stationary state with energy E is o 2 W d a 2 e 2 tanH0 2MZe z \