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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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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 38, Problema 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?
Guida verificata passo dopo passo1
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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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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07:19
Moles & Avogadro's Number
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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04:27Power of Pushing a Box
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