In quantum electrodynamics, the Uehling potential describes the interaction potential between two electric charges which, in addition to the classical Coulomb potential, contains an extra term responsible for the electric polarization of the vacuum. This potential was found by Edwin Albrecht Uehling in 1935. Uehling's corrections take into account that the electromagnetic field of a point charge does not act instantaneously at a distance, but rather it is an interaction that takes place via exchange particles, the photons. In quantum field theory, due to the uncertainty principle between energy and time, a single photon can briefly form a virtual particle–antiparticle pair, that influences the point charge. This effect is called vacuum polarization, because it makes the vacuum appear like a polarizable medium. By far the dominant contribution comes from the lightest charged elementary particle, the electron. The corrections by Uehling are negligible in everyday practice, but it allows the calculation of spectral lines of hydrogen-like atoms with high precision.
Definition The Uehling potential is given by (units c = 1 {\displaystyle c=1} and ℏ = 1 {\displaystyle \hbar =1} )
V ( r ) = − e 2 4 π r ( 1 + e 2 6 π 2 ∫ 1 ∞ d x e − 2 r m e x 2 x 2 + 1 2 x 4 x 2 − 1 ) , {\displaystyle V(r)={\frac {-e^{2}}{4\pi r}}\left(1+{\frac {e^{2}}{6\pi ^{2}}}\int _{1}^{\infty }dx\,e^{-2rm_{\text{e}}x}{\frac {2x^{2}+1}{2x^{4}}}{\sqrt {x^{2}-1}}\right),}
from where it is apparent that this potential is a refinement of the classical Coulomb potential. Here m e {\displaystyle m_{\text{e}}} is the electron mass and e {\displaystyle e} is the elementary charge measured at large distances. If r ≫ 1 / m e {\displaystyle r\gg 1/m_{\text{e}}} , this potential simplifies to
V ( r ) ≈ − e 2 4 π r ( 1 + e 2 16 π 3 / 2 1 ( m e r ) 3 / 2 e − 2 m e r ) , {\displaystyle V(r)\approx {\frac {-e^{2}}{4\pi r}}\left(1+{\frac {e^{2}}{16\pi ^{3/2}}}{\frac {1}{(m_{\text{e}}r)^{3/2}}}e^{-2m_{\text{e}}r}\right),}
while for r ≪ 1 / m e {\displaystyle r\ll 1/m_{\text{e}}} we have
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