Quantum Optics

Quantum Optics is a subfield of physics that applies the principles of quantum mechanics to study the fundamental nature of light and its interaction with matter. Moving beyond the classical wave description, this field treats light as being composed of discrete, quantized particles called photons, allowing for the investigation of phenomena that have no classical counterpart, such as quantum entanglement, superposition, and the particle-like behavior of individual photons. The principles of quantum optics are foundational to the operation of lasers and are driving the development of next-generation technologies including quantum computing, quantum cryptography, and ultra-precise sensors.

1. Foundations of Classical and Quantum Physics
   1. Classical Electrodynamics
      1. Maxwell's Equations
         1. Gauss's Law for Electric Fields
         2. Gauss's Law for Magnetic Fields
         3. Faraday's Law of Electromagnetic Induction
         4. Ampère-Maxwell Law
      2. Electromagnetic Wave Propagation
         1. Wave Equation Derivation
         2. Plane Wave Solutions
         3. Spherical and Cylindrical Waves
      3. Energy and Momentum in Electromagnetic Fields
         1. Poynting Vector
         2. Energy Density
         3. Radiation Pressure
      4. Boundary Conditions for Electromagnetic Fields
         1. Dielectric Interfaces
         2. Conducting Surfaces
      5. Classical Harmonic Oscillator
         1. Simple Harmonic Motion
         2. Driven Oscillator
         3. Damped Oscillator
         4. Energy and Phase Relations
      6. Polarization of Electromagnetic Waves
         1. Linear Polarization
         2. Circular Polarization
         3. Elliptical Polarization
         4. Jones Vector Formalism
         5. Stokes Parameters
         6. Poincaré Sphere
   2. Early Quantum Theory
      1. Blackbody Radiation
         1. Classical Rayleigh-Jeans Law
         2. Ultraviolet Catastrophe
         3. Planck's Quantum Hypothesis
         4. Planck Distribution
      2. Photoelectric Effect
         1. Experimental Observations
         2. Einstein's Photon Theory
         3. Work Function
         4. Threshold Frequency
         5. Stopping Potential
      3. Compton Scattering
         1. Photon-Electron Collisions
         2. Energy and Momentum Conservation
         3. Compton Wavelength
      4. Wave-Particle Duality
         1. de Broglie Matter Waves
         2. Double-Slit Experiments
         3. Complementarity Principle
   3. Quantum Mechanical Foundations
      1. Postulates of Quantum Mechanics
         1. State Vectors and Hilbert Space
         2. Observables and Operators
         3. Measurement Postulate
         4. Time Evolution
      2. Mathematical Framework
         1. Dirac Notation
         2. Inner Products and Orthogonality
         3. Completeness Relations
         4. Basis Sets
      3. Schrödinger Equation
         1. Time-Dependent Form
         2. Time-Independent Form
         3. Probability Current
      4. Operators and Observables
         1. Hermitian Operators
         2. Eigenvalue Problems
         3. Commutation Relations
         4. Uncertainty Principle
      5. Quantum Harmonic Oscillator
         1. Energy Eigenvalue Problem
         2. Ladder Operators
         3. Creation and Annihilation Operators
         4. Number States
         5. Zero-Point Energy
      6. Two-Level Quantum Systems
         1. Spin-1/2 Systems
         2. Bloch Sphere Representation
         3. Pauli Matrices
         4. Time Evolution
         5. Rabi Oscillations