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\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/

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# A A surprise result using Helmann-type potential

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