Lecture on Quantum Mechanics and the Principle of Least Action

Introduction

• Misconception: Belief that every object has a single trajectory
• Reality: Objects explore all possible paths simultaneously

Thought Experiment

• Scenario: Helping a friend in water
• Different paths:
  • Shortest path: Directly towards friend, involves more swimming
  • Alternative path: Running along the beach, longer total distance
  • Optimal path: Depends on running and swimming speeds
• Related to light and medium: Light takes the fastest path, similar to humans choosing optimal path

Misconception of Trajectories

• Initial belief: Light and objects follow a single path
• Reality: Light explores all possible paths, similar to quantum particles like electrons and protons
• Nature's illusion: We perceive single, well-defined trajectories

Concept of Action

• Maupertuis: Introduced concept of 'action' as mass x velocity x distance
• Hamilton: Action is integral of (kinetic energy - potential energy)
• Action's significance in physics: Useful in solving problems when Newton's laws are cumbersome

Birth of Quantum Mechanics

• Historical Context: Electric lighting in Germany, 1890s
• Blackbody radiation: Objects emit radiation based on temperature
• Rayleigh-Jeans law: Predicted infinite energy at short wavelengths (UV catastrophe)
• Max Planck's solution:
  • Energy quantization: Energy comes in multiples of quantum (E=hf)
  • Planck's constant introduced as a quantum of action

Development of Quantum Theory

• Albert Einstein: Light as discrete packets (photons)
• Niels Bohr: Stability of atoms through quantized angular momentum
• Louis de Broglie: Matter particles as waves, leading to wave-particle duality

Feynman's Path Integral Formulation

• Every particle/path considers all possible paths
• Double-slit experiment analogy: Particles go through both slits
• Richard Feynman's insight: Infinite paths considered, even when traveling through space

Consequences of Path Integral

• Classical mechanics emerges from quantum mechanics
• Phase and action determine probability of paths
• Paths that constructively interfere are observed

Demonstration by Casper

• Light and paths reflected in various directions
• Proves light explores all paths, but destructive interference cancels most

Conclusion

• Principle of least action: Central to understanding physics
• Theoretical physicists focus on action, not energy or forces
• Lagrangian framework: Potential for unifying laws of physics
• Ongoing search for a theory of everything

Call to Action

• Invitation to Q&A on Patreon
• Acknowledgments to NordVPN for sponsorship