Analytical model for a hydrogen atom in a magnetic field: Implications for the diamagnetic shift

We propose a simple phenomenological model for a hydrogen atom in a magnetic field which correctly reproduces the results of Makado and McGill's variational calculations for the $s$, $p$, $d$, and $f$ states for orbital quantum numbers $n\le 4$. The expression for the energy of the $1s$ state reveals that it never exhibits true free carrier behavior. Even at high magnetic fields the energy varies as ${\alpha }_n\hslash {\omega }_c/2$, where $\hslash {\omega }_c=\hslash eB/{\mu }_r$ is the cyclotron energy and ${\alpha }_n\simeq 0.84<1$. A Taylor expansion of the expression for the energy of the $1s$ state predicts a diamagnetic shift $\Delta E=\sigma B^2$ where $\sigma ={\alpha }_n^2{e}^2{a}_B^2/4{\mu }_r$. This suggests that there is a missing factor of ${\alpha }_n^2\simeq 0.71$ in the generally accepted perturbation expression for the diamagnetic shift. This hypothesis is supported by a careful analysis of literature data for shallow donors and excitons in GaAs in the magnetic field.