Perturbations on The Energy Levels of Electron in The Penning Trap, Due to The Dirac Equation

The penning trap, invented in 1959 by H. Dehmelt, is a combination of an inner electric dipole field and a constant magnetic field. A charged particle trapped in the Penning trap is called a geonium atom. The energy levels of the geonium atom are known in the non-relativistic limit, using the solutions of the Schrodinger equation, and for the case when the gravitational field of the Earth is negligible. In this article we write the Dirac equation for the geonium atom, coupled to the weak gravitational field of the Earth, and study its energy levels using perturbation techniques. If the Newtonian gravitational potential is , the dimensionless perturbing parameter is  , where c is the velocity of light. It is found that the effect of these perturbations affects the energy levels by a factor of almost . This tiny shift in the energy levels may be used to investigate the effect of the gravitational field on earth.