Domanda del libro di testo
At what speed is an electron’s de Broglie wavelength (a) 1.0 nm, (b) 1.0 μm, and (c) 1.0 mm?
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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 17
INT Through what potential difference must an electron be accelerated from rest to have a de Broglie wavelength of 500 nm?
Guida verificata passo dopo passo1
Step 1: Recall the formula for the de Broglie wavelength, which is \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle.
Step 2: Relate the momentum \( p \) to the kinetic energy \( K \) of the electron using \( p = \sqrt{2m_e K} \), where \( m_e \) is the mass of the electron and \( K \) is the kinetic energy.
Step 3: The kinetic energy \( K \) of the electron is equal to the work done on it by the electric field, which is \( K = e \Delta V \), where \( e \) is the charge of the electron and \( \Delta V \) is the potential difference.
Step 4: Combine the equations \( \lambda = \frac{h}{\sqrt{2m_e e \Delta V}} \) to express \( \Delta V \) in terms of \( \lambda \), \( h \), \( m_e \), and \( e \). Rearrange to solve for \( \Delta V \): \( \Delta V = \frac{h^2}{2m_e e \lambda^2} \).
Step 5: Substitute the known values: \( h = 6.626 \times 10^{-34} \, \text{J·s} \), \( m_e = 9.109 \times 10^{-31} \, \text{kg} \), \( e = 1.602 \times 10^{-19} \, \text{C} \), and \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \) into the formula to calculate \( \Delta V \).
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de Broglie Wavelength
The de Broglie wavelength is a fundamental concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For an electron, this wavelength can be calculated when its velocity is known, linking wave-like and particle-like behavior.
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Percorso guidato
05:42
Unknown Wavelength of Laser through Double Slit
Kinetic Energy and Potential Difference
When an electron is accelerated through a potential difference (V), it gains kinetic energy equal to the work done on it by the electric field. This relationship is expressed as KE = eV, where KE is the kinetic energy, e is the charge of the electron, and V is the potential difference. This principle is crucial for determining the energy required to achieve a specific velocity, which in turn affects the de Broglie wavelength.
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07:24Potential Energy Graphs
Momentum of an Electron
The momentum of an electron is defined as the product of its mass and velocity (p = mv). In the context of quantum mechanics, the momentum is also related to the de Broglie wavelength. As the electron is accelerated, its velocity increases, leading to a change in momentum, which directly influences the wavelength according to the de Broglie relation.
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05:17Intro to Momentum
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