A 10.0-g marble is gently placed on a horizontal tabletop...

A $10.0-\mathrm{g}$ marble is gently placed on a horizontal tabletop that is 1.75 $\mathrm{m}$ wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 14 billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

Key Concepts

Quantum Effects in Macroscopic Objects

While quantum mechanical effects like the Heisenberg uncertainty principle theoretically affect all objects, their practical implications for macroscopic objects (such as a marble) are extremely small. In everyday contexts, these quantum uncertainties are negligible compared to classical uncertainties, yet they establish a fundamental limit on the precision with which any physical system can be described.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a core tenet of quantum mechanics that states there is a fundamental limit to the precision with which pairs of physical properties, such as position and momentum, can be simultaneously known. This limitation is not due to experimental imperfections, but rather is an intrinsic property of nature at the quantum scale.

Measurement and Uncertainty

Measurement uncertainty refers to the concept that any measurement has an inherent indeterminacy, often due to the limits of the measurement tools or the quantum nature of particles. In contexts like quantum mechanics, even if an object seems macroscopically well-defined, there is still an intrinsic uncertainty described by fundamental limits.

Momentum–Velocity Relationship

Momentum and velocity are directly linked through the relation p = mv, where m is the mass of the object. When uncertainty is present in momentum due to the Heisenberg principle, this uncertainty translates to an uncertainty in velocity. This relationship is critical when evaluating how unknown the motion of an object remains even if it appears at rest from a classical perspective.

Temporal Evolution of Uncertainty

The uncertainty in an object's momentum (and consequently its velocity) leads to a spread in its position over time. This temporal propagation of uncertainty is important because even a very small initial uncertainty in velocity can lead to a significant positional uncertainty over a long period, thereby linking measurement uncertainties to dynamical evolution.

Transcript

0:00 Okay.

00:02 In this problem, we have a marble on a table.

00:04 The marble has a mass of 10, of 0 .1 grams, rather, and the table is 1 .75 meters wide.

00:15 We're first asked to find the maximum uncertainty in its position.

00:20 From there, find the minimum uncertainty in its velocity.

00:24 And lastly, we're asked to find what's the uncertainty in the time it takes to fall off the table.

00:28 So for equations, this is clearly a heisenberg uncertainty principle problem.

00:34 And that uncertainty principle is that the product of uncertainties in x, position, and momentum is always going to be greater than or equal to a constant, h -bar.

00:44 And we can combine that with the definition of momentum, mv, to find uncertainties and velocity.

00:51 The only concept we're going to need to use for this problem is h -bar, which is h divided by 2 pi, and that's equal to 1 .055 times 10 to negative 34 joules seconds.

01:01 So let's jump into it.

01:05 So to start the uncertainty at x at its maximum is going to be the size of the table.

01:13 It can't be any larger.

01:14 The marble has to be somewhere on the table.

01:17 So we can state simply that delta x is 1 .75 meters.

01:24 Now to get delta vx, its velocity, uncertainty in velocity, we can take this uncertainty principle, divide delta x or the other, or rather combine with momentum...

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