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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 mathematical relationship between ω and k?
What is the expression for group velocity in terms of phase velocity and wavelength?
What is the physical meaning of group velocity?
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What is the relationship between k and λ?
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What is the expression for dk/dλ?
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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?
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What is the physical meaning of phase velocity?
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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.
