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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 16
At what speed is an electron’s de Broglie wavelength (a) 1.0 nm, (b) 1.0 μm, and (c) 1.0 mm?
Guida verificata passo dopo passo1
Step 1: Recall the de Broglie wavelength formula: λ = h / (m * v), where λ is the wavelength, h is Planck's constant (6.626 × 10⁻³⁴ J·s), m is the mass of the electron (9.109 × 10⁻³¹ kg), and v is the speed of the electron. Rearrange the formula to solve for v: v = h / (m * λ).
Step 2: For part (a), substitute λ = 1.0 nm = 1.0 × 10⁻⁹ m into the formula. Use the known values of h and m to calculate the speed v. Ensure all units are consistent (e.g., meters for wavelength, kilograms for mass).
Step 3: For part (b), substitute λ = 1.0 μm = 1.0 × 10⁻⁶ m into the formula. Again, use the same values for h and m to calculate the speed v. Follow the same unit consistency checks.
Step 4: For part (c), substitute λ = 1.0 mm = 1.0 × 10⁻³ m into the formula. Use the same values for h and m to calculate the speed v. Ensure unit consistency as before.
Step 5: After substituting the values for each case, simplify the expression for v in each scenario. The resulting speeds will correspond to the de Broglie wavelengths of 1.0 nm, 1.0 μm, and 1.0 mm, respectively.
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de Broglie Wavelength
The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like behavior of particles. 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 by relating its momentum to its speed and mass.
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05:42
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Momentum
Momentum is a vector quantity defined as the product of an object's mass and its velocity (p = mv). In the context of quantum mechanics, momentum plays a crucial role in determining the de Broglie wavelength of a particle. For an electron, as its speed increases, its momentum increases, leading to a shorter de Broglie wavelength.
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05:17Intro to Momentum
Planck's Constant
Planck's constant (h) is a fundamental constant in quantum mechanics, approximately equal to 6.626 x 10^-34 Js. It relates the energy of a photon to its frequency and is essential in the calculation of the de Broglie wavelength. This constant signifies the scale at which quantum effects become significant, influencing the behavior of particles like electrons.
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08:59Phase Constant of a Wave Function
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