  Contrary to popular belief, the Zeeman effect can be treated exactly in
single-electron systems, for arbitrary magnetic field strengths, as long as the
term quadratic in the magnetic field can be ignored. These formulas were
actually derived already around 1927 by Darwin, using the classical picture of
angular momentum, and presented in their proper quantum-mechanical form in 1933
by Bethe, although without any proof. The expressions have since been more or
less lost from the literature; instead, the conventional treatment nowadays is
to present only the approximations for weak and strong fields, respectively.
However, in fusion research and other plasma physics applications, the magnetic
fields applied to control the shape and position of the plasma span the entire
region from weak to strong fields, and there is a need for a unified treatment.
In this paper we present the detailed quantum-mechanical derivation of the
exact eigenenergies and eigenstates of hydrogen-like atoms and ions in a static
magnetic field. Notably, these formulas are not much more complicated than the
better-known approximations. Moreover, the derivation allows the value of the
electron spin gyromagnetic ratio $g_s$ to be different from 2. For
completeness, we then review the details of dipole transitions between two
hydrogenic levels, and calculate the corresponding Zeeman spectrum. The various
approximations made in the derivation are also discussed in details.
