Question Video: Calculating the Minimum Uncertainty of a Muon’s Momentum Physics • 9th Grade

A muon that is produced in a particle accelerator has an uncertainty in its position of 2.00 × 10⁻¹¹ m. Using the formula Δ𝑥Δ𝑝 ≥ ℎ/4𝜋, calculate the minimum possible uncertainty in the momentum of the muon. Use a value of 6.63 × 10⁻³⁴ J.s for the Planck constant. Give your answer to 3 significant figures.

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Video Transcript

A muon that is produced in a particle accelerator has an uncertainty in its position of 2.00 times 10 to the negative 11th meters. Using the formula Δ𝑥 times Δ𝑝 is greater than or equal to ℎ over four 𝜋, calculate the minimum possible uncertainty in the momentum of the muon. Use a value of 6.63 times 10 to the negative 34th joule-seconds for the Planck constant. Give your answer to three significant figures.

In this exercise, we have a muon, a subatomic particle that’s like an electron but more massive, moving through a particle accelerator. There’s some amount of uncertainty about exactly where this muon is. We say that’s the uncertainty in its position, and we call it Δ𝑥. Given the value for Δ𝑥, we want to calculate the minimum possible uncertainty in the muon’s momentum. And we’ll do it using this equation: Δ𝑥 times Δ𝑝 is greater than or equal to ℎ over four 𝜋.

Mathematically, this equation tells us that there’s a limit to how precisely we can know both position and the momentum of our object, in this case, a muon. That limit of precision is achieved when Δ𝑥 times Δ𝑝 is equal to Planck’s constant divided by four 𝜋. And since in this exercise, we’re solving for the minimum possible value of Δ𝑝, we’ll use that equality. We can begin solving for Δ𝑝 by multiplying both sides by one over Δ𝑥, canceling that out on the left. And that gives us this expression.

We know the value for ℎ. That’s given to us in the problem statement; that’s Planck’s constant. And we’re also told the uncertainty in the position of our muon, Δ𝑥. When we substitute in these values and calculate Δ𝑝, to three significant figures, we find a result of 2.64 times 10 to the negative 24th kilograms meters per second. That’s the minimum uncertainty in the muon’s momentum.

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