Video Transcript
An electron in a particle
accelerator has an uncertainty in its position of 5.11 times 10 to the negative 14th
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 electron. 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 example then, we have an
electron moving at some high speed in a particle accelerator. At any instant, this electron has
some position and it also has some amount of momentum. And we know its position to within
an uncertainty of 5.11 times 10 to the negative 14th meters. This means if we had some distance
like this, we could say that the electron is somewhere within that distance. In other words, that’s our
uncertainty in its position. And as we saw, that uncertainty is
given to us as this number.
Along with this, the electron has
some uncertainty in its momentum, that is, its mass times its velocity. We want to solve for this
uncertainty. And we’re going to do it using this
formula here: Δ𝑥 times Δ𝑝 is greater than or equal to ℎ over four 𝜋. In this equation, Δ𝑥 is the
uncertainty in our object’s position. Δ𝑝 is the uncertainty in its
momentum. ℎ is Planck’s constant. And this inequality here says that
Δ𝑥 times Δ𝑝 is always greater than or equal to this value here.
Now, our question specifically asks
us to calculate the minimum possible uncertainty in the momentum of the
electron. That minimum in Δ𝑝 occurs when Δ𝑥
times Δ𝑝 is equal to ℎ over four 𝜋. So, we’re going to rewrite this
equation with an equal sign, which means that we’re solving for the minimum possible
uncertainty for Δ𝑥 times Δ𝑝. Since we’re looking to solve for
Δ𝑝, let’s divide both sides of this equation by Δ𝑥, canceling that out on the
left. And so, the minimum uncertainty in
the momentum of the electron is Planck’s constant ℎ divided by four 𝜋 times
Δ𝑥.
When we plug in the given values
for Planck’s constant and Δ𝑥, the answer we calculate, to three significant
figures, is 1.03 times 10 to the negative 21st kilograms meters per second. This is the minimum possible
uncertainty in this electron’s momentum.
