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.
