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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
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 39, Problema 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?
Guida verificata passo dopo passo1
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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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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08:30
Intro to Wave Functions
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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08:30Intro to Wave Functions
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