Hydrogenic Atoms in a Magnetic Field

The interaction of hydrogenic atoms with a weak constant magnetic field is discussed in detail. The Breit Hamiltonian, minimally coupled to the external magnetic field, is treated in several different ways. First, approximate eigenfunctions are obtained in the nonrelativistic nucleus approximation. These wave functions are used to treat perturbatively the residual terms dependent on the magnetic field, and to identify the magnetic moment of the bound electron in the ground state. The corrections previously given by us, of relative order ${\left(Z\alpha \right)}^2$, $\frac{{\left(Z\alpha \right)}^{2}m}{M}$, $\alpha {\left(Z\alpha \right)}^2$, and $\frac{\alpha {\left(Z\alpha \right)}^{2}m}{M}$, are confirmed including lowest-order radiative corrections. Next, a unitary transformation of the complete Breit Hamiltonian is made in order to simplify further the calculation of small corrections to the electron and nuclear $g$ factors. The physical origin of this unitary transformation, which is similar to a gauge transformation, is discussed extensively, and it is shown that the transformed Hamiltonian for a neutral system commutes with $\overrightarrow{\mathrm{P}}$, the momentum conjugate to the center-of-mass position $\overrightarrow{\mathrm{X}}$. This new Hamiltonian, which treats the electron and the nucleus on equal footing, is then transformed by means of the Chraplyvy-Barker-Glover reduction. The electron and the nuclear $g$ factors are calculated, this time including terms of relative order $\frac{{\left(Z\alpha \right)}^2{m}^2}{M^2}$ and $\frac{\alpha {\left(Z\alpha \right)}^2{m}^2}{M^2}$. These computations yield the magnetic moments for hydrogenic atoms in their ground state. The theoretical results are summarized and compared with recent experiments.