Quantum Mechanics

1. Core Principles and Equations
   1. Schrödinger Equation
      1. Time-dependent Schrödinger equation
         1. Introduction to wave functions
         2. Evolution of quantum states over time
         3. Role in non-equilibrium systems
      2. Time-independent Schrödinger equation
         1. Stationary states and energy eigenvalues
         2. Applications to quantum wells and oscillators
         3. Solutions in potential wells and barriers
      3. Applications to bound states
         1. Quantum harmonic oscillator
         2. Hydrogen atom
         3. Energy quantization and spectral lines
   2. Quantum Operators
      1. Observables and eigenstates
         1. Definition of observable operators
         2. Eigenvalues as measurable quantities
         3. Measurement postulate of quantum mechanics
      2. Commutators
         1. Relationship with uncertainty principle
         2. Use in determining simultaneous observables
         3. Calculation examples with position and momentum
      3. Hermitian operators
         1. Properties and significance in quantum theory
         2. Role in ensuring real eigenvalues
         3. Application in quantum mechanical symmetries
   3. Bra-Ket Notation
      1. Dirac notation
         1. Introduction to bra and ket vectors
         2. Simplified expression of complex quantum states
         3. Use in quantum mechanics problems
      2. Inner and outer products
         1. Definition and calculation of inner products
         2. Outer products and their use in state projections
         3. Applications in quantum state transformations
   4. Quantum Tunneling
      1. Potential barrier problems
         1. Calculation of transmission and reflection coefficients
         2. Quantum mechanical implications of classically forbidden regions
         3. Examples in chemical and nuclear processes
      2. Tunneling in semiconductor devices
         1. Role in the operation of transistors and diodes
         2. Impact on device miniaturization
         3. Quantum mechanical explanation of electronic conductivity