Example 33.1 from College Physics, 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited

Calculating the Mass of a Pion

Taking the range of the strong nuclear force to be about 1 fermi ( $10^{- 15} m$ ), calculate the approximate mass of the pion carrying the force, assuming it moves at nearly the speed of light.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
The calculation is approximate because of the assumptions made about the range of the force and the speed of the pion, but also because a more accurate calculation would require the sophisticated mathematics of quantum mechanics. Here, we use the Heisenberg uncertainty principle in the simple form stated above, as developed in Probability: The Heisenberg Uncertainty Principle. First, we must calculate the time $Δ t$ that the pion exists, given that the distance it travels at nearly the speed of light is about 1 fermi. Then, the Heisenberg uncertainty principle can be solved for the energy $Δ E$ , and from that the mass of the pion can be determined. We will use the units of $MeV / c^{2}$ for mass, which are convenient since we are often considering converting mass to energy and vice versa.
Solution
The distance the pion travels is $d \approx c Δ t$ , and so the time during which it exists is approximately
$\begin{array}{l} Δ t & \approx & \frac{d}{c} = \frac{10^{- 15}}{3 . 0 \times 10^{8}} \\ \approx & 3.3 \times 10^{- 24} s. \end{array}$
33.2

Now, solving the Heisenberg uncertainty principle for $Δ E$ gives

$Δ E \approx \frac{h}{4 π Δ t} \approx \frac{6 . 63 \times 10^{- 34} J \cdot s}{4π (3 . 3 \times 10^{- 24} s)} .$
33.3

Solving this and converting the energy to MeV gives

$Δ E \approx (1 . 6 \times 10^{- 11} J) \frac{1}{1 . 6 \times 10^{- 13} J} = 100 .$
33.4

Mass is related to energy by $Δ E = {mc}^{2}$ , so that the mass of the pion is $m = Δ E / c^{2}$ , or

$m \approx 100 .$
33.5
Discussion
This is about 200 times the mass of an electron and about one-tenth the mass of a nucleon. No such particles were known at the time Yukawa made his bold proposal.

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