Textbook Question
At what speed is an electron’s de Broglie wavelength (a) 1.0 nm, (b) 1.0 μm, and (c) 1.0 mm?
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0. Math Review
Math Review
Problem 26a
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?
Verified step by step guidance1
Step 1: Understand the concept of probability in this context. The probability of finding the sphere's center at exactly x=50.0 mm is related to the fact that the sphere is moving back and forth between the walls in a perfectly elastic motion. Since the sphere is continuously moving, the probability of it being at an exact point (like x=50.0 mm) at a random instant is essentially zero. This is because the sphere spends no measurable time at any single point during its motion.
Step 2: Recognize that the sphere's motion is deterministic and periodic. The sphere moves back and forth between x=0 mm and x=100 mm, and its position at any given time can be described mathematically using its velocity and the time elapsed since the start of its motion.
Step 3: Consider the nature of continuous motion. In continuous motion, the sphere's position is described by a continuous function of time. The probability of the sphere being at an exact position (like x=50.0 mm) at a random instant is zero because the sphere does not 'pause' at any point—it is always moving.
Step 4: Reflect on the concept of probability density. While the probability of the sphere being at exactly x=50.0 mm is zero, you could calculate the probability density function for the sphere's position over the range of motion. This would give insight into the likelihood of the sphere being near x=50.0 mm, but not exactly at that point.
Step 5: Conclude that the probability of the sphere's center being at exactly x=50.0 mm at a random instant is zero due to the continuous nature of its motion. This is a fundamental property of continuous probability distributions, where the probability of being at a single point is zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Elastic Collisions
In physics, elastic collisions are interactions between two bodies where both momentum and kinetic energy are conserved. In this scenario, the sphere collides with the walls without losing speed, meaning that it reverses direction while maintaining its velocity. Understanding this concept is crucial for analyzing the motion of the sphere as it reflects off the walls.
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Intro To Elastic Collisions
Uniform Motion
Uniform motion refers to the movement of an object at a constant speed in a straight line. In this case, the sphere moves back and forth between two fixed points (the walls) at a constant speed. This concept is essential for determining the sphere's position over time and understanding its repetitive motion pattern.
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Probability in Continuous Systems
In continuous systems, the probability of finding an object at an exact position is theoretically zero due to the infinite number of possible positions. Instead, we often consider intervals or ranges. In this question, understanding how to apply probability to the sphere's position within the defined space is key to addressing the likelihood of it being at exactly x=50.0 mm.
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