YPH1001: Quantum Theory in Physics

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)