Ch 39: Wave Functions and Uncertainty

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 39: Wave Functions and Uncertainty

Problem 26b

Chapter 39, Problem 26b

A 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 mm. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is between x = 49.0 mm and x = 51.0 mm?

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The problem involves calculating the probability of the sphere's center being in a specific region during its motion. Since the motion is perfectly elastic and there is no loss of speed, the sphere moves back and forth uniformly between the walls. This means the probability distribution of the sphere's position is uniform across the range of motion.

The total range of motion of the sphere's center is from x = 0 mm to x = 100 mm. This gives a total range of 100 mm. The probability of the sphere's center being in any specific region is proportional to the length of that region divided by the total range.

The region of interest is between x = 49.0 mm and x = 51.0 mm. The length of this region is calculated as:

51.0 - 49.0

mm.

The probability of the sphere's center being in this region is given by the ratio of the length of the region to the total range of motion. This can be expressed as:

\frac{51.0 - 49.0}{100}

Simplify the expression to find the probability. This will give you the fraction of time the sphere's center spends in the specified region during its motion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Elastic Collisions

In elastic collisions, both momentum and kinetic energy are conserved. When the sphere collides with the walls, it reverses its direction without losing speed, maintaining its total energy. This principle is crucial for understanding the motion of the sphere as it bounces back and forth between the walls.

Uniform Motion

The sphere moves uniformly between the walls, meaning it travels at a constant speed. Since the collisions are perfectly elastic, the time taken to travel between the walls is consistent. This uniform motion allows us to analyze the probability of the sphere's position over a defined interval.

Probability Distribution

In this context, the probability distribution describes the likelihood of the sphere's center being in a specific range of positions. Given the uniform motion, the probability of finding the sphere in a small interval can be calculated based on the total distance traveled and the size of the interval, leading to a straightforward assessment of its position.

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Probability Distribution Graph

Related Practice

Textbook Question

FIGURE P39.28 shows a pulse train. The period of the pulse train is T = 2 Δt, where Δt is the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an electronics system with a 200 kHz bandwidth? (This is the bandwidth allotted to each FM radio station.)

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Textbook Question

What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.0 MHz oscillation?

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Textbook Question

Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model, draw A graph of |ψ(x)|² for the electrons on the detection screen.

110

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Textbook Question

An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval. If 10⁶ electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm?

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Textbook Question

A 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 mm. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is at exactly x = 50.0 mm?

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Textbook Question

What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3×10⁵ m/s and 4 ×10⁵ m/s? Give your answer to one significant figure.

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