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## Estimating Pion mass using the uncertainty principle

1. The problem is based uponYukawa's original prediction for pion mass-energy.

suppose the force between nucleons is due to the emission of a particle mass m from one nucleon and the absorption by another. given the range of the nuclear force is

R=(1.4)$$\times$$10$$^{-15}$$

use $$\Delta$$E$$\Delta$$t ~ h bar
to make an order of magnitude estimate of the mass energy in MeV

2
x=position
p=momentum
E=energy
m=pion rest mass
h bar = reduced planck constant
t=time
c=speed of light
$$\Delta$$=uncertainty in _

the equation required by the question is
$$\Delta$$E$$\Delta$$t ~ h bar

i also attempted using

E$$^{2}$$=p$$^{2}$$c$$^{2}$$ +m$$^{2}$$c$$^{4}$$

and the momentum form of the uncertainty principle

$$\Delta$$p$$\Delta$$x ~ h

I was pretty stumped because i couldn't fathom what the delta t represented, and how to use the required equation with just one variable. i used dimensional analysis as i knew it would have to be made up of constants h and c and the given value R (taken to be delta x) and got
E~$$\frac{hc}{R}$$
But this does not take into account any prefactors and doesn't use the required equation. i also tried subbing the momentum form of the uncertainty principle into thye relativistic energy equation, but to no avail as the Delta t is still there

This is one of the first questions from my introductory quantum mech. course, so i'm not particularly clue up on the physical principle, can someone please point me in the right direction?

Help!

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Mentor  Think of the speed of light as a conversion factor between length and time. You're given a length scale R; it'll correspond to a time scale, which gives you an estimate for $\Delta t$.  The uncertainty principle $\Delta E\Delta t \approx \hbar$ says that over short time scales, the energy of a system doesn't have a definite value but has a spread of values, so as long as a process, like exchanging a meson, occurs over a short enough period of time, the energy required for the process, like the energy of the meson, is available.

Tags
particle theory, pion, quantum, rest mass, uncertainty

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