Textbook Question
INT Through what potential difference must an electron be accelerated from rest to have a de Broglie wavelength of 500 nm?
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Ch 38: Quantization
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 38, Problem 21
What is the quantum number of an electron confined in a 3.0-nm-long one-dimensional box if the electron’s de Broglie wavelength is 1.0 nm?
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Understand the problem: The electron is confined in a one-dimensional box, which means we are dealing with the quantum mechanical particle-in-a-box model. The quantum number (n) determines the energy state of the electron, and we are given the de Broglie wavelength (λ) and the length of the box (L).
Recall the relationship between the de Broglie wavelength and the quantum number in a one-dimensional box: The allowed wavelengths are given by the formula: , where L is the length of the box and n is the quantum number.
Rearrange the formula to solve for the quantum number n: . This equation relates the quantum number to the box length and the de Broglie wavelength.
Substitute the given values into the formula: L = 3.0 nm and λ = 1.0 nm. Ensure the units are consistent (both are in nanometers, so no conversion is needed). The equation becomes: .
Interpret the result: The quantum number n must be a positive integer because it represents discrete energy levels. After performing the division, round the result to the nearest whole number if necessary to ensure it is an integer.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quantum Numbers
Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They include the principal quantum number, angular momentum quantum number, magnetic quantum number, and spin quantum number. In the context of a particle in a box, the quantum number indicates the energy level and the allowed states of the electron.
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de Broglie Wavelength
The de Broglie wavelength is a concept that relates the wavelength of a particle to its momentum, expressed as λ = h/p, where h is Planck's constant and p is the momentum. For an electron in a confined space, this wavelength can be used to determine the allowed energy states and corresponding quantum numbers. It highlights the wave-particle duality of matter.
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Particle in a Box Model
The particle in a box model is a fundamental quantum mechanics concept that describes a particle free to move in a small space with infinitely high potential walls. This model allows for the calculation of quantized energy levels and wave functions. The length of the box and the quantum number determine the energy levels and the corresponding wavelengths of the particle.
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Related Practice
Textbook Question
The diameter of the nucleus is about 10 fm. What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm?
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Textbook Question
The diameter of the nucleus is about 10 fm. A simple model of the nucleus is that protons and neutrons are confined within a one-dimensional box of length 10 fm. What are the first three energy levels, in MeV, for a proton in such a box?
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Textbook Question
The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. Draw the atom’s energy-level diagram. Label each level with the energy and the quantum number.
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Textbook Question
The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. An electron traveling with a speed of 1.30×106 m/s collides with the atom. Can the electron excite the atom to the n = 2 stationary state? The n = 3 stationary state? Explain.
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Textbook Question
What is the de Broglie wavelength of a 200 g baseball with a speed of 30 m/s?
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