Let us calculate numeric values for the Uehling potential in hydrogen, so that we get a grasp what is the magnitude of the effect.
Alexei M. Frolov and David M. Wardlaw (2012) found an analytic formula for the Uehling potential. They use Hartree atomic units.
Frolov and Wardlaw mention that the exponential decrease for the Uehling potential for large r is wrong when corrections to the electric field are taken into account.
The square of the 2s wave function very close to the nucleus is 1 / 2, if we use the Bohr radius as the unit of length.
Let us calculate the Uehling potential for a radius of the reduced Compton wavelength
r = 4 * 10⁻¹³ m
of the electron.
The formula (1) in the Frolov and Wardlaw link is easy to integrate in your head. The integral for r is very roughly
1 / (5 * 137) * 1 / 40 = 1 / 30,000
The Coulomb potential at r is
V = -511 keV / 137 = -4 keV.
The Uehling correction to the potential is
ΔV = 4 keV / 30,000 = -0.13 eV.
The volume of a sphere with the radius r is
Vol = 1.3 * 10⁻⁶,
where the unit is the Bohr radius.
The probability of the 2s electron to be in that sphere is
P = 1 / 2 * Vol = 0.7 * 10⁻⁶.
The fall in the energy level of 2s is
ΔE = P ΔV
= 0.09 μeV.
In literature, the effect of vacuum polarization is given as 27 MHz or 0.1 μeV.
The contribution from a radius ~ r / 10 is roughly -0.01 μ. The contribution from r / 100 is very small.
We conclude that the Uehling potential lowers the energy of 2s, and the main contribution comes from around a radius which is the reduced Compton wavelength.
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