Question involving Uncertainty Principle.

[brat-sampson]

Using the uncertainty principle for energy and time, estimate the range of the Strong interaction if it is due to the exchange of a pion with rest mass 140GeV (Take the value of h as 4.14x10^-21MeV s)
Now I know that Delta T x Delta E is > or = to hbar
and that I want DeltaX as a solution. But after having worked out DeltaT using basic algebra it's hard to know where to go. Should I head with E = 1/2 mv^2 to get the velocity then use speed x time to get distance? If so how do I find the mass of a pion? Or I could try and get the momentum but encounter similar problems.
If there's a simple way using the wave function then I might not know it as I really wasn't sure about that lecture and am working on finding out more.
Any help appreciated, thanks.

 

[Chi Meson]

pions have a lifespan of 2.6x10^-8 seconds. Does that help?

 

[mezarashi]

Your lecturer is probably asking you to derive something similar to what Yukawa originally postulated regarding the strong force. Your approach seems to be backwards however. At that time, they didn't know the mass of this once non-existant particle. The answer can be verified, as this 'range' is approximatey the size of atomic nuclei. If we assume that the pion can travel near the speed of light, then the time it would take to exchange would be:

$$\Delta t = \frac{r}{c}$$

For some r (range) you don't know. You can then relate this to an uncertainty in energy using the uncertainty principle. Subsequently, relate this to an uncertainty in mass using what else other than Einstein's famous $$E = mc^2$$