Analytical formulas for the third-order diamagnetic energy of a hydrogen atom

Abstract

Analytical expressions are presented for the third-order diamagnetic corrections to the energy of nondegenerate hydrogen levels with arbitrary principal quantum number $n$ and the magnetic quantum number $|m|=n\ensuremath{-}1,n\ensuremath{-}2,n\ensuremath{-}3$. The leading term for the third-order energy correction for levels with high $n$ is determined to be $\ensuremath{\Delta}{E}^{(3)}\ensuremath{\approx}\frac{3}{128}{n}^{16}{B}^{6}$. Together with the well-known first- and second-order corrections $\ensuremath{\Delta}{E}^{(1)}\ensuremath{\approx}\frac{1}{8}{n}^{4}{B}^{2}$ and $\ensuremath{\Delta}{E}^{(2)}\ensuremath{\approx}\ensuremath{-}\frac{1}{32}{n}^{10}{B}^{4}$ it determines the upper and lower bounds for the level energy in field and also the range of magnetic fields where the first- and second-order perturbation theory terms are valid for calculating the Zeeman energy in hydrogenlike states of atoms.