Imagine another universe in which the value of Planck's...

Imagine another universe in which the value of Planck's constant is 0.0663 $\mathrm{J} \cdot \mathrm{s}$ , but in which the physical laws and all other physical constants are the same as in our universe. In this universe, two physics students are playing catch. They are 12 $\mathrm{m}$ apart, and one throws a 0.25 -kg ball directly toward the other with a speed of 6.0 $\mathrm{m} / \mathrm{s} .$ (a) What is the uncertainty in the ball's horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a cube with volume 125 $\mathrm{cm}^{3}$ at the time she throws it? (b) By what horizontal distance could the ball miss the second student?

Key Concepts

Heisenberg's Uncertainty Principle

This principle in quantum mechanics states that there is a fundamental limit to the precision with which certain pairs of physical properties—such as the position and momentum of a particle—can be simultaneously known. It implies that a more precise measurement of one quantity will inherently lead to a less precise measurement of the conjugate quantity, and the product of the uncertainties is bounded from below by a value determined by Planck's constant.

Planck's Constant

Planck's constant is a fundamental physical constant that sets the scale of quantum effects. It appears in many areas of quantum mechanics and determines the size of the quantum of action. In the context of the uncertainty principle, it provides the numerical lower bound for the product of the uncertainties in position and momentum.

Quantum Uncertainty in Macroscopic Contexts

While quantum uncertainty is typically significant in microscopic systems, its principles also apply to macroscopic objects. Altering the value of Planck's constant would modify the scale at which these uncertainties become noticeable. This concept underscores how quantum mechanical limits are universal, even though their observable effects may vary dramatically between microscopic and everyday scales.

Transcript

00:01 Welcome to digital tea with mr.

00:03 E.

00:03 Where we see what's brewing in the world of physics education.

00:07 Today we're going to take a look at a quantum mechanics style problem, algebra -based.

00:14 This one deals with the heisenberg uncertainty principle.

00:17 The gist of the problem is that we are to imagine that we are in another universe, where planck's constant is a far larger number at 0 .663, uh, jewels, uh, jules seconds.

00:32 And we got two students having a catch with a ball.

00:35 The students are 12 meters apart.

00:37 Uh, the ball has a mass of 0 .25 kilograms.

00:41 Uh, if quantum effects were not being considered, the ball would be moving at, uh, we would know for sure the ball is moving at six meters per second.

00:50 Um, however, this, with the plunks constant being so, uh, large, uh, the person that's receiving the ball, understands that the ball's initially starting in a 125 centimeter cube.

01:08 And they know that with certainty, although the fact that there's a cube there leads to the uncertainty.

01:15 Basically, this is heisenberg's uncertainty principle, in that you can know either the momentum or position of an object, but the more you know one of them, you know less of the other one.

01:28 So the key to start with this solution is that box of 125 cubic centimeters.

01:34 If we take the cube root of that, that would give us dimensions of 5 centimeters by 5 by 5 centimeters.

01:50 And if i consider that the box is completely centered around the head of the person that's going to be receiving it, this distance across the whole top would be 5 centimeters.

02:08 So how i'm going to use that in my problem is i'm going to divide that by 2, meaning the ball could end up landing 2 .5 centimeters in front or behind.

02:19 So you have 2 .5 centimeters and then put that in the metric.

02:23 That would be the delta x of 0 .025 meters for the, uncertainty of the i guess the original starting position of the ball so now with that as an acceptable uncertainty we're going to then see what the correlated uncertainty in the momentum is equal to so we can express that as a change in momentum is equal to plonks constant over four pi times the uncertainty in the position substituting our numbers in, we would have 0 .663, oh, 63, my bad.

03:16 Put a zero up there.

03:18 0663 over 4 pi times 0 .25.

03:26 Run that through the calculator, and that gives us an uncertainty and momentum of 0211.

03:36 Kilogram meters per second.

03:40 So now, that would be the first part of the question, the uncertainty and the momentum.

03:46 The second part of the question wants to know by how much could the ball miss the target by, where the target's 12 meters away.

03:55 The approach for that one i looked at is if i take a look at this uncertainty and momentum, and i consider it equal to the standard momentum equation of mass times velocity, or in this case delta velocity, i want to find the uncertainty in the velocity.

04:14 So that would be a delta v equals delta p over m.

04:18 So run the numbers through that.

04:21 You got our 0 .211 over 0 .25 was it? yeah, 0 .25 kilograms...

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