Solid State Physics

1. Theoretical Foundations
   1. Quantum Mechanics in Solid State
      1. Wave-Particle Duality
         1. Particle-like and wave-like behaviors
         2. Relevance to electron behavior in solids
      2. Schrödinger Equation
         1. Time-dependent and time-independent forms
         2. Applications in potential wells and barriers
      3. Quantum Tunneling
         1. Tunneling in semiconductors
         2. Role in modern devices like tunnel diodes
      4. Uncertainty Principle
         1. Position and momentum uncertainties
         2. Implications for electron behavior in solids
   2. Bloch's Theorem
      1. Periodic Potentials
         1. Mathematical description of periodic potentials in crystals
         2. Applicability to electron wavefunctions
      2. Bloch Functions
         1. Definition and properties
         2. Significance in electronic band structure
      3. Applications of Bloch’s Theorem
         1. Prediction of energy bands
         2. Influence on electrical and thermal properties
   3. Density Functional Theory (DFT)
      1. Thomas-Fermi Model
         1. Early model of electron distribution
         2. Limitations and developments towards DFT
      2. Hohenberg-Kohn Theorems
         1. Fundamental theorems underpinning DFT
         2. Concepts of electron density and energy functionals
      3. Kohn-Sham Equations
         1. Derivation and application in DFT
         2. Exchange-correlation functional approximations
      4. Practical Applications of DFT
         1. Material property predictions
         2. Use in computational materials science
   4. Fundamental Concepts in Solid State Theory
      1. Lattice Dynamics
         1. Harmonic and anharmonic oscillations
         2. Impact on thermal properties and phonon interactions
      2. Electron-Phonon Interactions
         1. Mechanisms and implications
         2. Role in superconductivity and electrical resistance
      3. Spin and Magnetism Theories
         1. Quantum mechanical treatment of spins
         2. Exchange interactions and magnetic ordering
      4. Many-Body Problem
         1. Concepts of electron-electron interactions
         2. Simplifications and solutions, like Hartree-Fock method
   5. Advanced Computational Methods
      1. Ab Initio Methods
         1. Non-empirical methods and their implications
         2. Examples: Quantum Monte Carlo, GW approximation
      2. Tight-Binding Model
         1. Principles and application in band structure calculations
         2. Comparison with ab initio approaches
      3. Molecular Dynamics
         1. Simulation of atomic and molecular interactions
         2. Application to phase transitions and material properties
   6. Recent Developments and Challenges
      1. Multiscale Modeling
         1. Bridging different scales from atomic to macroscopic
         2. Integrated approaches combining DFT with classical methods
      2. Quantum Computing in Solid-State Physics
         1. Algorithm development for solid-state simulations
         2. Quantum annealing and its potential for solving complex problems
      3. Limitations of Current Theories
         1. Accuracy and computational complexity
         2. Challenges in predicting properties of novel materials