The Coherent State of the Landau Hamiltonian and the Relativistic Corrections to the Zeeman Effect in He⁺ Ions

Abstract

We calculate the energy levels of He + ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He + ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He + ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h , j 2 , l 2 , s 2 and j z , where j , l and s are the internal total, orbital and spin angular moments of the He + ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He + ion. In addition to the Z 2 a 2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy ( E CM ) of the CM circular motion of the He + ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He + ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He + ions in magnetized astrophysical objects.