Hydrogen atom in a strong magnetic field

We study the energy spectrum of atomic hydrogen in strong ($B > B_{\rm a}\sim 10^9$ G) and ultra-strong ($B \gtrsim B_{\rm cr} \sim 10^{14}$ G) magnetic fields, in which the hydrogen electron starts to move relativistically and quantum electrodynamics effects become important. Within the adiabatic approximation, highly accurate energy level values are obtained analytically for $B > 10^{11}$ G, which are then compared with asymptotic and numerical results available in the literature. A characteristic feature noted in electron motion in a strong magnetic field is that for $B \gtrsim B_{\rm cr}$, the transverse motion becomes relativistic, while the longitudinal motion (along В) can be described by nonrelativistic theory and is amenable to the adiabatic approximation. Topics discussed include: the qualitative difference in the way odd and even levels change with the magnetic field (for $B \gg B_{\rm a}$); the removal of degeneracy between odd and even atomic states; spectral scaling relations for different quantum numbers $(n, n_\rho, m)$ and different field strengths; the shape, size, and quadrupole moment of the atom for $B \gg B_{\rm a}$; radiative transitions $n{\rm p\to 1s}$ in a strong magnetic field; relativistic QED effects, including the effects of vacuum polarization and of the electron anomalous magnetic moment on the energy level positions; Coulomb potential screening and energy level freezing at $B\to \infty $; and the possibility of the Zeldovich effect in the hydrogen spectrum in a strong magnetic field. The critical nuclear charge problem is briefly discussed. Simple asymptotic formulas for $Z_{\rm cr}$, valid for low-lying levels, are proposed. Some of the available information on extreme magnetic fields produced in the laboratory and occurring in space is given. The Coulomb renormalization of the scattering length is considered in the resonance situation with a shallow level in the spectrum.