One way to estimate the range of an interaction is by using the Heisenberg uncertainty principle. Assume that the uncertainty in the energy of an exchange particle is given by its rest energy and that the particle travels at nearly the speed of light. Under those assumptions, what is the approximate range of the weak interaction mediated by the Z⁰ particle, which has a mass of 91.2 GeV/c²? A) 10⁻¹⁵ m B) 10⁻¹⁸ m C) 10⁻²¹ m D) 10⁻²⁴ m E) 10⁻²⁷ m

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Quizplus · Accepted Answer

The range of an interaction can be estimated using the Heisenberg uncertainty principle as follows: ΔxΔp ~ ℏ where Δx is the range of the interaction, Δp is the uncertainty in momentum of the exchange particle, and ℏ is the reduced Planck constant. Assuming that the exchange particle has a rest energy of 91.2 GeV/c² and travels at nearly the speed of light, its momentum can be approximated as: p ~ E/c ~ 91.2 GeV/c The uncertainty in momentum can be approximated by the uncertainty in energy, which is given by the rest energy of the particle: Δp ~ 91.2 GeV/c Substituting these values into the uncertainty principle equation, we get: Δx ~ ℏ/Δp ~ (6.6 × 10^-34 J s)/(91.2 × 10^9 eV/c) ~ 7.2 × 10^-18 m Converting the units to the desired format, we get: Δx ~ 10^-18 m = 10^-18 × 10^12 ^{− 6} m = 10^-18+12 − 6 ⁻ m = 10^-18+6 × (10^-12) ⁻ m = 10^-18+4 ⁻ m Therefore, the approximate range of the weak interaction mediated by the Z⁰ particle is 10⁻¹⁸ meters, corresponding to choice (B).

One Way to Estimate the Range of an Interaction Is