The strong nuclear interaction has a range of approximately 0.70 × 10−¹⁵ m. It is thought that an elementary particle is exchanged between the protons and neutrons, leading to an attractive force. Utilize the uncertainty principle to estimate the mass of the elementary particle if it moves at nearly the speed of light. A) 140 MeV/c², a pion B) 511 keV/c², an electron C) 96 GeV/c², a Z boson D) 500 MeV/c², a K meson E) 106 MeV/c², a muon

Question

Quizplus · Accepted Answer

According to the uncertainty principle, the uncertainty in the momentum of the particle ($\Delta p$) multiplied by the uncertainty in its position ($\Delta x$) is approximately equal to Planck's constant ($\hbar$):
$$\Delta p \Delta x \approx \hbar$$
If we assume that the elementary particle moves at nearly the speed of light, we can use the relationship between momentum and energy to estimate its mass. Specifically, the energy of a particle with momentum $p$ and mass $m$ is given by:
$$E^2 = (pc)^2 + (mc^2)^2$$
At nearly the speed of light, we can neglect the mass component and approximate this as:
$$E \approx pc$$
Combining this with the uncertainty principle, we have:
$$\Delta p \Delta x \approx \frac{\hbar}{c}$$
Solving for $\Delta p$ and setting $\Delta x$ equal to the range of the strong nuclear interaction, we get:
$$\Delta p \approx \frac{\hbar}{c \Delta x} \approx 200 	ext{ MeV/c}$$
This suggests that the mass of the exchanged particle is probably on the order of a few hundred MeV/c$^2$. Of the given choices, A is the only one in this range. Specifically, a pion has a mass of about 140 MeV/c$^2$, which is consistent with our estimate.

The Strong Nuclear Interaction Has a Range of Approximately 0