Nonrelativistic Electron in the Coulomb Field

In the first approximation, energy levels of one-electron atoms (see Fig. 1.1) are described by the solutions of the Schrodinger equation for an electron in the field of an infinitely heavy Coulomb center with charge Z in terms of the proton charge  

Eides et al. Theory of Light Hydrogenic Bound States, STMP 222, 1—12 (2007) 

Straightforward calculation of the characteristic values of the velocity, Coulomb potential and kinetic energy in the stationary states gives 

We see that due to the smallness of the fine structure constant a a one-electron atom is a loosely bound nonrelativistic system and all relativistic effects may be treated as perturbations. There are three characteristic scales 

We are interested only in low-Z atoms. High-Z atoms cannot be treated as nonrelativistic systems, since an expansion in Za is problematic. 

---