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 19
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?
Verified step by step guidance1
Determine the relationship between the bandwidth (Δf) and the time duration (Δt) of the pulse using the time-bandwidth product. For a Gaussian pulse, the time-bandwidth product is approximately Δt * Δf ≈ 0.44. Rearrange this to find the shortest pulse duration: Δt = 0.44 / Δf.
Substitute the given bandwidth of the optical fiber, Δf = 2.0 GHz (2.0 × 10⁹ Hz), into the equation for Δt to calculate the shortest pulse duration.
Calculate the period of one oscillation of the laser light using the relationship T = 1 / f, where f is the frequency of the laser. The frequency of the laser can be found using the speed of light equation: f = c / λ, where c = 3.0 × 10⁸ m/s (speed of light) and λ = 1.5 μm (1.5 × 10⁻⁶ m).
Substitute the calculated frequency of the laser into the equation T = 1 / f to find the period of one oscillation.
Determine the number of oscillations in the shortest-duration pulse by dividing the pulse duration (Δt) by the period of one oscillation (T): Number of oscillations = Δt / T.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Wavelength and Frequency
Wavelength and frequency are inversely related properties of waves. The wavelength (λ) is the distance between successive peaks of a wave, while frequency (f) is the number of oscillations per second, measured in hertz (Hz). For light, the speed of light (c) relates these two by the equation c = λf. Understanding this relationship is crucial for determining the number of oscillations in a laser pulse.
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Pulse Duration
Pulse duration refers to the time interval during which a pulse is active or has significant amplitude. In the context of laser pulses, shorter durations correspond to higher frequencies of oscillation. The duration of a pulse can be estimated using the bandwidth of the optical fiber, as a larger bandwidth allows for shorter pulse durations, which is essential for calculating the number of oscillations in the pulse.
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Bandwidth and Information Transmission
Bandwidth is the range of frequencies that a communication channel can transmit, measured in hertz. In optical fibers, a higher bandwidth allows for the transmission of more data and shorter pulse durations. The relationship between bandwidth and pulse duration is given by the time-bandwidth product, which indicates that a pulse's duration is inversely proportional to the bandwidth, thus affecting the number of oscillations in the pulse.
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Related Practice
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
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
FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the value of a?
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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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