Textbook Question
FIGURE EX39.16 shows the wave function of an electron. Draw a graph of |ψ(x)|2.
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 14c
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?
Verified step by step guidance1
Step 1: Understand the problem. The graph of |ψ(x)|² represents the probability density function for the electron's position. To find the probability that the electron is located between x = 1.0 nm and x = 2.0 nm, we need to integrate |ψ(x)|² over this interval.
Step 2: Write the integral expression for the probability. The probability P is given by: . This integral calculates the area under the curve of |ψ(x)|² between x = 1.0 nm and x = 2.0 nm.
Step 3: Analyze the graph of |ψ(x)|² provided in FIGURE EX39.14. Identify the functional form of |ψ(x)|² or approximate the values if the graph is discrete or irregular. If the graph provides a specific function, substitute it into the integral.
Step 4: Perform the integration. Depending on the functional form of |ψ(x)|², use appropriate integration techniques (e.g., analytical integration for simple functions or numerical methods for complex or discrete data). Ensure the limits of integration are x = 1.0 nm and x = 2.0 nm.
Step 5: Interpret the result of the integration. The value obtained from the integral represents the probability that the electron is located between x = 1.0 nm and x = 2.0 nm. Ensure the result is consistent with the physical interpretation of probabilities (i.e., it should be a number between 0 and 1).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Wave Function and Probability Density
In quantum mechanics, the wave function, denoted as ψ(x), describes the quantum state of a particle. The square of the absolute value of the wave function, |ψ(x)|^2, represents the probability density, indicating the likelihood of finding the particle at a specific position. This concept is fundamental for understanding how quantum particles behave and how their locations can be predicted.
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Integration of Probability Density
To find the probability of a particle being located within a specific range, one must integrate the probability density over that range. For the given question, this involves calculating the integral of |ψ(x)|^2 from x=1.0 nm to x=2.0 nm. This process quantifies the total probability of finding the electron in the specified interval, which is a key aspect of quantum mechanics.
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Normalization of the Wave Function
The wave function must be normalized, meaning that the total probability of finding the particle in all space equals one. This ensures that the probabilities derived from |ψ(x)|^2 are meaningful. Normalization is crucial for accurate calculations in quantum mechanics, as it establishes a consistent framework for interpreting probabilities associated with particle locations.
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Related Practice
Textbook Question
What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.0 MHz oscillation?
Textbook Question
FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at x = 0.000 mm?
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?
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
FIGURE EX39.12 shows the probability density for an electron that has passed through an experimental apparatus. If 1.0×106 electrons are used, what is the expected number that will land in a 0.010-mm-wide strip at 2.000 mm?