Potential Energy of Relativistic Particles in Coulomb Field

[reterty]

Let us consider relativistic particle (electron) which moves with relativistic speed $v$ in the Coulomb field (in the field of a fixed heavy nucleus). The main question is what is the potential energy of a particle in such a static field? Landau and Lifshitz in their book "Field Theory" believe that the potential energy is not renormalized in any way and is equal to $\frac{qQ}{r}$. At the same time, a number of authors of original articles on this topic introduce a reduced distance $r\sqrt{1-v^2/c^2}$ into the denominator of this fraction due to the relativistic effect of the reduction in linear dimensions. Which of them is right?

 

[vanhees71]

I guess you mean the relativistic motion of a charged particle in the coulomb field of a very much heavier particle, neglecting the radiation reaction. The relativistic equation of motion in the non-covariant formalism is derived from the Lagrangian
$$L=-mc^2 \sqrt{1-\dot{\vec{x}}^2} + \frac{q Q}{4 \pi \epsilon_0 |\vec{x}|}.$$
It's of some historical interest since it was Sommerfeld's derivation of the fine structure of the hydrogen-atom spectrum within old quantum theory. It's kind of surprising that he got the correct result although the model is, of course, entirely wrong, i.e., it doesn't take into account the spin 1/2 of the electron and the gyrofactor 2 (both of which weren't known in 1916). That's why you find the solution in Wikipedia here:

https://en.wikipedia.org/wiki/Bohr–Sommerfeld_model#Relativistic_orbit

 

[DrGreg]

$r\sqrt{1-v^2/c^2}$

Please note that on this website you need to use a double-$ instead of a single-$ for LaTeX to work.
$$r\sqrt{1-v^2/c^2}$$

 

[PeterDonis]

on this website you need to use a double-$ instead of a single-$ for LaTeX to work

Or a double # for inline LaTeX (the double $ means an equation in its own paragraph).

 

[reterty]

For example: https://www.researchgate.net/publication/305345527_A_New_Relativistic_Extension_of_the_Harmonic_Oscillator_Satisfying_an_Isochronicity_Principle