An approximately relativistic theory of bound states which ensures Poincaré invariance of the atomic system to relative order (v/c${)}^2$ developed by one of us [K. J. Sebastian, Phys. Rev. A 23, 2810 (1981)] is used to derive the Zeeman interaction Hamiltonian for a two-body system. This Hamiltonian is correct to order α(Zα${)}^2$ and to all orders of ${\mathit{m}}_{\mathit{e}}$/${\mathit{m}}_{\mathit{N}}$ and includes contributions from previously neglected radiative and recoil corrections. Explicit analytic expressions for the ${\mathit{g}}_{\mathit{J}}$ and ${\mathit{g}}_{\mathit{I}}$ factors are given and verify the results of Grotch and Kashuba [Phys. Rev. A 7, 78 (1973)]. This interaction Hamiltonian in conjunction with the zero-field relativistic interaction correct to order α(Zα${)}^4$ is used to analyze the Zeeman structure of the n=1, 2, and 3 levels of hydrogen and positronium. In the case of hydrogen, corrections for the Lamb shift which include the most recent radiative recoil and pure recoil corrections have also been included to yield an extremely precise analysis for hydrogen in the ‖${\mathit{nFm}}_{\mathit{F}}$JL〉 representation. The Hamiltonian, g factors, and Zeeman splittings are compared with previously obtained results and experimental results. The results of these comparisons are discussed at length.

• Received 11 June 1993

DOI:https://doi.org/10.1103/PhysRevA.49.192