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What is the probability of finding a 1s hydrogen electron at distance r > aB from the proton?
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Ch 41: Atomic Physics
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 41, Problema 44b
Suppose you put five electrons into a 0.50-nm-wide one-dimensional rigid box (i.e., an infinite potential well). What is the ground-state energy—that is, the total energy of all five electrons in the ground-state configuration?
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Understand the problem: We are dealing with a one-dimensional infinite potential well (rigid box) of width 0.50 nm. The task is to calculate the total ground-state energy of five electrons, considering the Pauli exclusion principle and the quantized energy levels of the system.
Recall the energy levels for a particle in a one-dimensional infinite potential well: The energy of the nth quantum state is given by the formula: , where is Planck's constant, is the mass of the particle (electron in this case), and is the width of the box.
Apply the Pauli exclusion principle: Electrons are fermions, so no two electrons can occupy the same quantum state. Each energy level can hold two electrons (one with spin up and one with spin down). For five electrons, the first two will occupy the n=1 level, the next two will occupy the n=2 level, and the fifth electron will occupy the n=3 level.
Calculate the energy for each occupied level: Use the formula for to calculate the energy of the n=1, n=2, and n=3 levels. Multiply the energy of each level by the number of electrons occupying that level (2 for n=1, 2 for n=2, and 1 for n=3).
Sum the energies: Add the contributions from all occupied levels to find the total ground-state energy of the system. This will give you the total energy of the five electrons in the ground-state configuration.
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Quantum Mechanics and Infinite Potential Well
In quantum mechanics, an infinite potential well is a model where a particle is confined to a specific region of space with infinitely high potential barriers. This means the particle cannot exist outside this region. The energy levels of particles in such a well are quantized, leading to discrete energy states that depend on the width of the well and the mass of the particle.
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07:24
Potential Energy Graphs
Pauli Exclusion Principle
The Pauli Exclusion Principle states that no two fermions, such as electrons, can occupy the same quantum state simultaneously. In the context of the infinite potential well, this principle dictates that each electron must occupy a unique energy level, which affects the total energy calculation for multiple electrons in the system.
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14:47Diffraction with Huygen's Principle
Energy Levels in a Quantum Well
The energy levels of a particle in a one-dimensional infinite potential well are given by the formula E_n = n^2 * (h^2 / (8mL^2)), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the width of the well. For multiple electrons, the total ground-state energy is the sum of the energies of the lowest occupied levels, taking into account the Pauli Exclusion Principle.
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04:10Intro to Energy & Types of Energy
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During what interval of time will 10% of a sample of 2p hydrogen atoms decay?
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