# Rescaling the Uehling potential formula

I found that the Uehling potential equation is written in natural units in many books and research papers. But because I’ve to make some calculations with the Uehling correction, the formula must be written in non-natural scales. The whole Uehling potential I found here (SciHub) is given as

$$U(x)= \frac{-e^2}{4π ϵ_0 x}(1+\frac{2α}{3π}∫_0^∞(\frac{1}{z^2}+\frac{1}{2z^4})exp⁡(\frac{-2zx}{λ ̅_C})\sqrt{z^2-1}dz)$$

Whereas, $$λ ̅_C$$ is the reduced Compton wavelength, and $$α$$ is the fine-structure constant. In the same paper i found, the above formula is rewritten as,

$$U(x)=\frac{-e^2}{4π ϵ_0 x}+δ(x)$$

One of the numerical solutions of $$δ(x)$$ in the research paper (in paper 2 in the research paper i mentioned here ), and the solution I want rescale is,

$$δ(x)=-\frac{2αe^2}{12π^2ϵ_0 x}(-γ_E-\frac{5}{6}+ln⁡(\frac{1}{cx}))$$

The above formula is valid for small values of $$x$$ only (shorter than reduced Compton wavelength),$$γ_E$$ is the Euler-Mascheroni constant.

I forgot how to rescale (and, honestly, I was bad at natural units and stuff like that!! because i have bad memory). So, could some one kind please help me out, thank you guys.

• Related question by OP: physics.stackexchange.com/q/494336/2451 – Qmechanic Aug 5 '19 at 11:30
• That was a question i deleted long time ago because it was not edited well @Qmechanic – MichaelPhysica Aug 5 '19 at 12:04
• The recommended procedure for improving a question on SE is to edit the question within the same thread rather than starting new threads. – Qmechanic Aug 5 '19 at 12:24
• thank you very much for your recommendations @Qmechanic. – MichaelPhysica Aug 5 '19 at 13:00
• Your $\delta(x)$ is missing the prefactor $e^2/(4\pi\epsilon_0x)$. – G. Smith Aug 5 '19 at 16:04