A surprise result using Helmann-type potential

[ftr]

\begin{align}
V(r,a)=\frac{e^2}{r}-\frac{e^2}{r}exp(\frac{-r}{a})
\end{align}

The above equation is called Helmann type potential which is a combination of Yukawa and coulomb potential. It is used to solve many problems in physics, for example
https://arxiv.org/pdf/1307.2983.pdf

But I noticed a remarkable property, which could be just some coincidence. Now, with Yukawa potential we typically interpret "a" in above equation as the exchange mass for the force. However here we interpret "a" just as length for inverse of two masses that INTERACT with each other.

if we use e^2=3 and use several values for "a"(the horizontal column in Exel sheet) and plot against distance, we see something strange. It seems that all the curves converge to a value of .000272287.

I used 300,700,1300,1711 for the shown plot. but you can use any such numbers. Also I have calculated for 1836 as an info only.

so if we multiply .000272287 by 2 and take the inverse we get 1836.2977 which is close to the proton-electron mass ratio. Or in another way, the electron mass can be taken to be .000272287*2=0.0005445

Now we figure which r,a will give us 1 which is the mass of the proton compared to the electron mass.
by inspection I find a nice solution(although others exist) r=3.8, a=3.8 that gives the potential to be
.499 and multiply that by 2 you get almost 1
If we take 1836.1526 to be the electron reduced Compton wave and see the value of 3.8 by comparison
we have 3.86159e-13*3.8/1836.1527= .7991738 e-15

which is very close to proton radius

The system is also remarkable in that it is scale invariant

I don't know what all that means. but I wonder if it is possible to get this potential from first principle at least.

You can use this to do and verify all calculations

3/r-(3/r)*EXP(-1*r/a)
http://m.wolframalpha.com/

 

[mfb]

The curves go to zero. If you evaluate them at a large r/a ratios, then you just get approximately 3/r. For r=11000, this gives 3/11000, if you invert it and divide it by 2 you get 11000/6 = 1833. That is close to the electron to proton mass ratio. So what? The number 11000 is completely arbitrary. Pick r=6000 and the same calculation gives 1000. Pick r=6000 pi and the same calculation gives 1000 pi.

 

[ftr]

The curves go to zero. If you evaluate them at a large r/a ratios, then you just get approximately 3/r. For r=11000, this gives 3/11000, if you invert it and divide it by 2 you get 11000/6 = 1833. That is close to the electron to proton mass ratio. So what? The number 11000 is completely arbitrary. Pick r=6000 and the same calculation gives 1000. Pick r=6000 pi and the same calculation gives 1000 pi.

Yes, it is somewhat arbitrary. However, given the fact if you zoom that 11000 is the START of "good" convergence and it coincides with an interesting number it seems to be saying something, coupled to the proton results.

But yes, there must be a way to confirm that position independently. I think I know how to get to it approximately, but unfortunately I am not feeling well now , maybe a bit later.

 

[mfb]

11000 is not the start of "good convergence". It is the point where the diagram with your chosen values doesn't show differences any more. Zoom in or take a=10000 and it will show differences. Consider a smaller range for a and you'll get curves that are very close together earlier. There is absolutely nothing special about 11000, and we don't do numerology here. I closed the thread.