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, j2, l2, s2 and jz, 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 Z2 a2 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 (ECM ) 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.
Highlights
When a composite system with a nonzero net electric charge is placed in a uniform magnetic field, neither the total canonical or mechanical momenta are conserved
In a recent paper [9] we have shown that the quantum state which most closely resembles the classical state of a charged particle moving in a uniform magnetic field, is the coherent state of the Landau quantum Hamiltonian of such a particle
The main goal of this paper is to calculate the corrections to the eigenvalues or energy levels of h(0) + h(1) of the He+ ion due to the presence of an external uniform magnetic field, the so called Zeeman effect on a composite system with a net electric charge
Summary
When a composite system with a nonzero net electric charge is placed in a uniform magnetic field, neither the total canonical or mechanical momenta are conserved. There is no inertial frame where the total momentum is zero at all times. The problem of separating the c.m. motion from the internal motion for such a system will be different from that of an isolated system where the total or the c.m. momentum is conserved. Several authors [1]-[8] have studied the problem of calculating the corrections to the internal energy levels of composite ions in a magnetic field. Some of them [1] [2] [3] [4] are based on a constant of motion introduced initially by Baye.
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