Range of a mediating particle

Question

My book describes the derivation of the formula $R\approx\hbar/2mc$ by: $$(\Delta E)(\Delta t)\geq\hbar/2$$ The violation of energy conservation is $\Delta E=mc^2$ to create the particle’s mass. Also, the particle travels at relativistic speed, so $R=c\Delta t$. $$(mc^2)(R/c)\approx\hbar/2$$ $$R\approx\hbar/2mc^2$$

However, why is the energy needed to make the particle $mc^2$? The particle is not at rest, so don’t we need to include the kinetic term? Would $E^2=p^2c^2+m^2c^4$ not be a more suitable equation to substitute into Heisenberg’s Uncertainty Principle?

This treatment is completely wrong, see this question for a discussion of how to use the energy-time uncertaintly relation. The range of a mediating particle is given by its mass by inspecting the corresponding Yukawa potential, not by some uncertainty relation. — ACuriousMind, Nov 05 '15 at 13:44

score 3 · Accepted Answer · answered Nov 05 '15 at 17:53

As ACuriousMind says in a comment, this isn't the approach Yukawa used in his 1935 paper (Yukawa H 1935 Proc. Phys. Math. Soc. Japan 17 48) though whether he did that calculation in the privacy of his own notebook only he knows.

The calculation you describe is a rather arm waving sort of justification for the relationship between the mass of the mediating particle and the range, but it shouldn't be taken too literally. You're quite correct that we should be including the kinetic energy not just the rest mass energy, but really this is such a vague calculation that it doesn't make any difference. At best you should regard the calculation as an order of magnitude estimate.

Range of a mediating particle

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