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Questions and Answers
What is the relationship between phase velocity and group velocity when the component waves move more slowly than the envelope?
What is the relationship between phase velocity and group velocity when the component waves move more slowly than the envelope?
What is the mathematical relationship between ω and k?
What is the mathematical relationship between ω and k?
What is the expression for group velocity in terms of phase velocity and wavelength?
What is the expression for group velocity in terms of phase velocity and wavelength?
What is the physical meaning of group velocity?
What is the physical meaning of group velocity?
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What is the relationship between k and λ?
What is the relationship between k and λ?
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What is the expression for dk/dλ?
What is the expression for dk/dλ?
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What happens to the component waves when phase velocity equals zero?
What happens to the component waves when phase velocity equals zero?
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What is the expression for v_g in terms of v_p and λ?
What is the expression for v_g in terms of v_p and λ?
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What is the physical meaning of phase velocity?
What is the physical meaning of phase velocity?
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What is the relationship between group velocity and phase velocity?
What is the relationship between group velocity and phase velocity?
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Study Notes
Module II: Quantum Theory
- Covers topics such as black body radiation spectrum, Wien's law, Rayleigh-Jeans law, quantum theory of radiation, wave mechanics, wave-particle duality, de Broglie waves, Bohr's quantization rules, phase and group velocities, Heisenberg Uncertainty Principle, wave function, and Schrodinger's wave equation.
Wave Particle Duality
- Radiant energy shows wave and particle nature, and similarly, matter possesses the same dual nature under suitable conditions.
- The wave-particle duality was first predicted by Louis de Broglie in 1924 and experimentally verified by Davission and Germer in 1927 and Thomson in the same year.
De Broglie Waves
- The wavelength of the associated wave is λ = h/p = h/mv, where m is the mass and v is the velocity.
- Example: Calculate the de Broglie wavelength associated with a 50 eV electron, λ = 1.66 Å.
Wave and Particle Nature
- Our traditional understanding of a particle: "Localized" - definite position, momentum, confined in space.
- Our traditional understanding of a wave: "de-localized" - spread out in space and time.
- Oscillation at a particular point: ψ = A cos 2πt = A cos ωt.
Wave Function and Wave Packet
- A wave packet is a group of waves with slightly different wavelengths interfering with each other in a way that the amplitude of the group (envelope) is non-zero only in the neighborhood of the particle.
- A wave packet is localized, a good representation for a particle.
Phase and Group Velocities
- Phase velocity: The rate at which the phase of the wave propagates in space.
- Group velocity: The rate at which the envelope of the wave packet propagates.
- The group velocity is the speed of the wave packet, and the phase velocity is the speed of the individual waves.
- Relations between phase velocity and group velocity:
- vg = vp - λ(dvp/dλ)
- vg = vp + k(dvp/dk)
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Description
Quiz on Quantum Theory, covering Black body Radiation spectrum and Wien's law, part of Physics course YPH1001.
