Textbook Question
FIGURE EX39.16 shows the wave function of an electron. Draw a graph of |ψ(x)|2.
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Ch 39: Wave Functions and Uncertainty
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 39, Problem 21
What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.0 MHz oscillation?
Verified step by step guidance1
Understand the relationship between the duration of a pulse and the bandwidth required to transmit it. The uncertainty principle in signal processing states that the product of the time duration of a signal (Δt) and its bandwidth (Δf) is approximately constant: Δt * Δf ≈ 1.
Determine the time duration of the pulse. Since the pulse consists of 100 cycles of a 1.0 MHz oscillation, calculate the total time duration (Δt) of the pulse using the formula: Δt = (number of cycles) / (frequency). Here, the frequency is 1.0 MHz, and the number of cycles is 100.
Rearrange the uncertainty principle formula to solve for the bandwidth (Δf): Δf ≈ 1 / Δt. Substitute the value of Δt obtained in the previous step into this formula.
Perform unit conversions if necessary. Ensure that the frequency and time duration are in compatible units (e.g., seconds for time and Hz for frequency) before calculating the bandwidth.
Interpret the result. The calculated bandwidth (Δf) represents the minimum frequency range required to transmit the pulse without significant distortion.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bandwidth
Bandwidth refers to the range of frequencies that a transmission medium can carry. In the context of signal transmission, it is crucial for determining how much data can be sent over a channel in a given time. A higher bandwidth allows for the transmission of more information, while a lower bandwidth restricts the amount of data that can be transmitted.
Pulse Transmission
Pulse transmission involves sending a signal that varies in amplitude or frequency over time, typically representing data. The characteristics of the pulse, such as its duration and frequency, influence the bandwidth required for effective transmission. Understanding the relationship between pulse shape and bandwidth is essential for optimizing communication systems.
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Example 1
Nyquist Theorem
The Nyquist Theorem states that to accurately transmit a signal without distortion, the sampling rate must be at least twice the highest frequency present in the signal. This principle is fundamental in determining the minimum bandwidth required for transmitting a pulse. For a signal with a frequency of 1.0 MHz, the minimum bandwidth needed would be at least 2.0 MHz to ensure proper transmission of the pulse.
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Related Practice
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 between x = 49.0 mm and x = 51.0 mm?
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Textbook Question
FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the probability that the electron is located between x = 1.0 nm and x = 2.0 nm?
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Textbook Question
A 1.5-μm-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?
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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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