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Problem 35 Hard Difficulty

In the ground state of hydrogen, the uncertainty in the position of the electron is roughly 0.10 nm. If the speed of the electron is approximately the same as the uncertainty in its speed, about how fast is it moving?

Answer

5.8 \times 10^{5} \mathrm{m} / \mathrm{s}

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Top Physics 103 Educators
Elyse G.

Cornell University

Marshall S.

University of Washington

Farnaz M.

Other Schools

Meghan M.

McMaster University

Video Transcript

Okay, So for this one, we know that the uncertainty of the position of the electron is point when the Anna meter is so I'm gonna call this Delta X this 0.1 nanometers, um, and then we want to get the uncertainty and the speed. So what we can do is use the heist Heisenberg uncertainty principle. And then, um, probably explains why in the text. But we could just say that we can change the greater than or equal, then signed to, like, an equal sign. Um, it's probably beyond the scope of this course to really do anything beyond that. Um, so it's equal to age bar over too. Is that what it is? Actually? Sorry, I'm just gonna double check. I've always had trouble remembering the exact formula for that here. Not such a basic, um, fact, but wow. Whatever. So maybe I should pause the video. I'm gonna go ahead and pasta video. I do this. I I was, uh I was right. Um, I'll go ahead and write it just like the book, right? So, just to be clear, a ce possible. So that's each divided by for pie and P. Let's assume that the masses has no uncertainty to it. Then Delta X, Delta P is gonna be am Delta Bi. So go ahead and put the AM out here and say that Sam Delta Bi is h over for pie. And then if you want to solve for Delta V, we can just divide both sides by this, um, Delta X. So a Delta V is equal to age over for pie divided by, um um Delta X. And so, if I can plug all of this into a calculator, each other for pyre one over and out, the ex yours, H uh, I was won over 9.109 cent in the minus 31 Built axes. What? Wasn't 0.1 times 10 to the minus nine. But really, what x was It's double trick that point when the enemy in areas in Great with that, I got that. The uncertainty is, which is? They're saying the problem. You could say it's equal. This tree is 5.79 times 10 of the 12345123455 100.79 times out of the five meters per second. May I say That sounds pretty high. Um, indeed, you generally need relativistic corrections for this behavior of the electron in an atom, um, into it because it is sort of closest view of life.

University of Washington
Top Physics 103 Educators
Elyse G.

Cornell University

Marshall S.

University of Washington

Farnaz M.

Other Schools

Meghan M.

McMaster University