Quadratic Zeeman effect for hydrogen: A method for rigorous bound-state error estimates

We present a variational method, based on direct minimization of energy, for the calculation of eigenvalues and eigenfunctions of a hydrogen atom in a strong uniform magnetic field in the framework of the nonrelativistic theory (quadratic Zeeman effect). Using semiparabolic coordinates and a harmonic-oscillator basis, we show that it is possible to give rigorous error estimates for both eigenvalues and eigenfunctions by applying some results of Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)]. The method can be applied in this simple form only to the lowest level of given angular momentum and parity, but it is also possible to apply it to any excited state by using the standard Rayleigh-Ritz diagonalization method. However, due to the particular basis, the method is expected to be more effective, the weaker the field and the smaller the excitation energy, while the results of Kato we have employed lead to good estimates only when the level spacing is not too small. We present a numerical application to the ${\mathit{m}}^{\mathit{p}}$=$0^{+}$ ground state and the lowest ${\mathit{m}}^{\mathit{p}}$=$1^{\mathrm{-}}$ excited state, giving results that are among the most accurate in the literature for magnetic fields up to about ${10}^{10}$ G.