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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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TOC.tableOfContents
- 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
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Capitolo 39, Problema 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?
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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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Concetti chiave
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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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05:08
Circumference, Period, and Frequency in UCM
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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04:49Example 1
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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05:32Introduction to Polarization
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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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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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