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≤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 αₙωc/2, where ωc=eB/μᵣ is the cyclotron energy and αₙ0.84<1. A Taylor expansion of the expression for the energy of the $1s$ state predicts a diamagnetic shift ΔE=σB² where σ=αₙ²e²aB²/4μᵣ. This suggests that there is a missing factor of αₙ²0.71 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.