Quantum Physics Lecture - October 22, 2024

Today's Agenda

• Topic: Quantum Physics
• Lecture: 18, October 22nd, 2024

Learning Objectives

• Write and calculate De Broglie wavelength for given particles
• Understand when particle wave nature is evident
• Derive time-dependent and time-independent Schrödinger equations in one dimension, extend to 3D
• Explain when to use time-dependent vs. time-independent Schrödinger equations
• Explain the physical significance of wave function of a particle

Historical Perspective

• 1921: Stern-Gerlach experiment, space quantization of angular momentum and spin
• 1923: Compton Scattering
• 1924: De Broglie's hypothesis, matter waves
• 1925: Heisenberg's matrix formulation of quantum mechanics
• 1926: Erwin Schrödinger's wave formulation of quantum mechanics
• 1926: Einstein-Bohr debates
• 1927: Davisson-Germer experiment, J.J Thomson
• 1941: Amplitude formulation by Feynman

De Broglie's Wavelength

• Any particle with mass (m) and velocity (v) can be associated with a wave
• Energy of a moving particle = energy of its associated wave
• De Broglie wavelength (λ_dB) = h / p = h / (mv) where:
• h = Planck's constant

Explanation for Quantization of Angular Momentum

• Electron waves visualized in Bohr orbits
• Orbiting electron as a standing wave around nucleus
• Angular momentum (mvr) is quantized: mvr = nh / 2π (n = integer)

Wave Nature - Particle Nature

• Wave and particle aspects are complementary
• Wave nature introduces uncertainty in particle location
• Wave nature dominates when wavelength is comparable to or larger than the measurement scale

Wave Packet

• Wave packet is a group of waves
• Combining different wavelengths and phases creates a localized wave packet
• Represents both wave and particle nature

Phase Velocity & Group Velocity

• Phase velocity doesn't represent the particle speed/rate of phase movement
• Group velocity is the rate at which the envelope propagates
• Group velocity equals particle velocity for matter waves

Matter Waves

• Matter waves exist only when the particle is in motion
• All particles (charged and uncharged) have matter waves
• Particle mass and wavelength are inversely related (heavier particles have smaller wavelengths)
• Group velocity is the particle velocity
• Phase velocity is greater than the speed of light
• Wave-particle duality principle: Only One aspect is observed at a time

Case Study: De Broglie Wavelength for an Electron

• Kinetic energy of an electron moving with velocity (v) expressed in terms of potential (V)
• Equation for De Broglie wavelength for an electron -λ = h / √(2meV), where m is the electron mass, e is the electron charge, V is the potential, and h is Planck's constant

Test your understanding of quantum physics with this quiz focusing on key concepts such as De Broglie wavelength and Schrödinger equations. Challenge yourself on the historical milestones and their significance in the development of quantum mechanics. Ideal for students looking to deepen their knowledge in this fascinating subject.