Two electron relativistic corrections to PE

[qsa]

What is the relativitic correction to the e^2/r coulomb law without spin between two electrons.second order is enough.

 

[dextercioby]

Second order in what ? Standard references mention a correction proportional to the 4th power of momentum, if the electron's and the proton's spin are neglected.

 

[qsa]

Second order in what ? Standard references mention a correction proportional to the 4th power of momentum, if the electron's and the proton's spin are neglected.

I guess I wanted the next most significant correction term, also I want electron-electron not electron-proton(which is usually given as hydrogen atom). Can you please post some links to the standerd references that you mentioned.

 

[Vanadium 50]

Why would electron-electron have a different functional form than electron-proton?

 

[qsa]

Why would electron-electron have a different functional form than electron-proton?

of course they shouln't but in case of hydrogen the system is in a bound state, I am not interested at such complication(if any) at this time.

 

[Bill_K]

What is the relativistic correction to the e^2/r coulomb law?

To the next order, V(r) = -(Ze²/4π)1/r - (Ze⁴/60π²m²) δ(r)

I know the delta function looks weird, like it was something just stuck in by hand. But that's really the result. It looks more sensible in momentum space, where

V(k) ~ k^-2 (1 - (e²/60π²m²) k² + ...)

and the Fourier transform of the second term is the Fourier transform of 1, which is a delta function.

 

[unusualname]

I think you're looking for something like this qsa

Effective Field Theory of Gravity: Leading Quantum Gravitational Corrections to Newtons and Coulombs Law

where the first order correction is shown to be an additional

3G(m1+m2)/(r*c^2)

(multiplied by the classical coloumb term)

(obviously m1=m2 for the electron, and r is the separation)

But, personally, I would give up running naive random models in the hope of getting physical laws, you'll go crazy. If your new model matches this formula it's still not a big deal, especially not if you don't explain how it's constrained in a coherent and simple manner.

 

[qsa]

To the next order, V(r) = -(Ze²/4π)1/r - (Ze⁴/60π²m²) δ(r)

I know the delta function looks weird, like it was something just stuck in by hand. But that's really the result. It looks more sensible in momentum space, where

V(k) ~ k^-2 (1 - (e²/60π²m²) k² + ...)

and the Fourier transform of the second term is the Fourier transform of 1, which is a delta function.

Thank you. what is the order of magnitude of the extra term (lets say for Z=1)

(Ze⁴/60π²m²) δ(r)

is there any experimental confirmation. Do you have any links.

 

[qsa]

I think you're looking for something like this qsa

Effective Field Theory of Gravity: Leading Quantum Gravitational Corrections to Newtons and Coulombs Law

where the first order correction is shown to be an additional

3G(m1+m2)/(r*c^2)

(multiplied by the classical coloumb term)

(obviously m1=m2 for the electron, and r is the separation)

But, personally, I would give up running naive random models in the hope of getting physical laws, you'll go crazy. If your new model matches this formula it's still not a big deal, especially not if you don't explain how it's constrained in a coherent and simple manner.

This is another story for another time since these corrections are out of reach of experiment. Bill_K gave the correct answer. As for my model ,you know I cannot talk about it here, I will send you an email soon with the latest results(maybe alpha up to eight digits).

 

[Bill_K]

The reference for the expression I quoted is Bjorken and Drell, Vol I, Sect 8.2 on Vacuum Polarization, especially Eq 8.27. The delta function in the second term limits its effect to S states, since they are the only ones whose wavefunction does not vanish at the origin. This term is partially responsible for the Lamb shift, however the contribution to the Lamb shift from other effects is larger.

 

[unusualname]

This is another story for another time since these corrections are out of reach of experiment. Bill_K gave the correct answer. As for my model ,you know I cannot talk about it here, I will send you an email soon with the latest results(maybe alpha up to eight digits).

The paper says the corrections due to quantum gravity ( ~ Gh/(r^2c^3) ) are out of experimental reach, the relativistic correction is ~ Gm/(rc^2) (it summarises this in the summary).

But the paper talks about scalar (spin 0) particles (which is what you asked for) so wouldn't directly apply to a realistic situation (it mentions that there is an extension to spin 1/2 fermions)

I guess you just wanted radiative correction from qed (eg Landau & Lifgarbagez, Vol 4 QED 2nd Ed p 504).

Maybe if you get more decimal places you can try to match it with something more state of the art, good luck!

 

[qsa]

Thanks to both of you. I wonder if this is from Breit or just Dirac. can anybody else help, I appreciate it? Is there a relation between bound and not bound states?