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How is group velocity (V_G) different from phase velocity?
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Group velocity (V_G) is defined as `dω/dk` and it represents the velocity of the wave packet or group of waves, whereas the phase velocity is the velocity at which any single phase of the wave travels.
Define the phase velocity (V_P) of a wave.
Phase velocity (V_P) is given by `V_P = ω/k`.
What phenomenon does classical mechanics fail to explain, known as the UV catastrophe?
Black Body Radiation
What does the photoelectric effect demonstrate about quantum mechanics?
The photoelectric effect demonstrates that light can behave as particles called photons, with energy quantized in units of `hv`, explaining phenomena that classical mechanics cannot.
What does the term 'wave function solutions' refer to in the context of a particle in a 1D box?
Wave function solutions refer to the possible forms of wave functions, denoted `ψ(x) = A sin(kx)` where `k = nπ/L`, which describe the state of the particle inside the box.
Describe the setup and potential function for a particle in a 1D box.
The setup consists of a particle confined in a box with boundaries at `0` and `L`, and the potential function is `V(x) = 0` inside the box and `∞` outside.
What are the energy eigenvalues for a particle in a 1D box?
The energy eigenvalues are `E_n = n² * π² * ħ² / (2mL²)` where `n = 1, 2, 3, ...`.
State Planck’s theory in resolving issues of classical physics.
Planck's theory introduces the concept of quantized energy levels to address inconsistencies in classical physics.
Provide the time-independent Schrodinger wave equation.
The time-independent Schrodinger equation is: `[-ħ²/2m * (d²ψ/dx²)] + V(x)ψ = Eψ`.
What is the physical interpretation of the wave function in quantum mechanics?
The wave function is used to calculate physical quantities like probabilities, energy, and momentum.
Explain De Broglie's concept of matter waves.
De Broglie proposed that matter, such as particles, can exhibit wave-like properties, introducing the concept of wave-particle duality.
What is the relation between group velocity and phase velocity in a non-dispersive medium?
In a non-dispersive medium, group velocity (V_G) is equal to phase velocity (V_P).
Derive the Compton wavelength shift equation.
The Compton wavelength shift (Δλ) is given by: `Δλ = h/m_ec * (1 - cos θ)` where `h` is Planck's constant, `m_e` is the electron mass, `c` is the speed of light, and `θ` is the scattering angle.
What does the term 'eigenfunctions' mean in quantum mechanics?
Eigenfunctions are solutions to the Schrodinger equation that correspond to specific, quantized energy levels known as eigenvalues.
Which experiment demonstrates the wave-particle duality, challenging classical mechanics?
The Double Slit Experiment
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