[QUOTE=bjschaeffer;1579003]The Yukawa potential is a solution of the Klein-Gordon equation
Assuming U independent of time, it may be written
where is the Compton wavelength of the particle. The simplest solution is then :
However, this solution is not physical: it is wrong to suppress the time derivative. We have to solve the complete equation with a periodical but stationary potential:
The complete equation is:
Replacing U we get after simplyfying the time dependent exponential :
The solution is almost the same as above :
There is a physical solution if :
That is when the solution is imaginary and tending to zero at infinity:[/QUOTE]
[QUOTE=ophase;1550210]Here's the original proof by Yukawa
Yukawa potential gs: Yukawa constant
Yukawa proposed that nuclear force has to be like elektromagnetic force. So the potential above need to satisfy second green equation with a source term:
Yukawa generalized the equation for non-static states.
This equation is also relativistical invariant. Then Yukawa quantized the potential:
Now we put that potential expression in the second green equation (*) and we get:
Here the last term should be the mass term: If we assume a=2 fm, then the exchange particle mass is mu= 100 MeV.
In 1947 Pion discovered at 140 MeV and it's proved that there is no meson in the nucleus according to Yukawa theory.
I don't know the rest of that story. Probably someone made a correction about the calculation above. Any ideas??[/QUOTE]