Quantum 101

Table of Contents Introduction What Is a Matrix? Matrix Operations Recap Matrix Representation of Linear Systems Matrices in Classical Mechanics Matrices in Electrodynamics Rotation Matrices and Coordinate Transformations Inertia Tensor in Rigid...

Table of Contents Introduction What Are Eigenvalues and Eigenvectors? The Characteristic Equation Computing Eigenvalues and Eigenvectors Geometric Interpretation Diagonalization and Similarity Transformations Defective Matrices and Jordan Form Applications of Eigenvalue Problems Spectral Theorem Hermitian...

Table of Contents Introduction What Is a Vector Space? Axioms of Vector Spaces Subspaces and Spanning Sets Linear Independence and Basis Dimension and Coordinate Systems Inner Product: Definition and Properties Examples of...

Table of Contents Introduction Scalars, Vectors, and Vector Spaces Linear Combinations and Span Linear Independence and Basis Matrices and Matrix Operations Linear Transformations The Rank of a Matrix Systems of Linear Equations...

Table of Contents Introduction What Is the Calculus of Variations? Functionals and Their Extremization The Euler-Lagrange Equation Derivation of the Euler-Lagrange Equation Boundary Conditions Examples of Variational Problems Variational Principles in Physics Lagrangian...

Table of Contents Introduction Motivation and Applications of Tensor Calculus Scalars, Vectors, and Tensors: A Unified Language Index Notation and Einstein Summation Convention Covariant and Contravariant Components Transformation Laws for...

Table of Contents Introduction Flat vs Curved Spacetime Curvature in General Relativity The Metric Tensor and Line Element Geodesics: The Straightest Paths Deriving the Geodesic Equation Christoffel Symbols and Covariant Derivatives Geodesics...

Table of Contents Introduction The Limitations of Special Relativity Equivalence Principle From Newtonian Gravity to Geometry Spacetime Curvature Tensors and the Metric Tensor Einstein’s Field Equations Geodesics and Motion in Curved Spacetime Gravitational...

Table of Contents Introduction Classical vs Relativistic Dynamics The Need for Redefining Momentum Four-Momentum and the Relativistic Framework Relativistic Momentum Relativistic Energy Energy-Momentum Relation Mass-Energy Equivalence Invariant Mass and Rest Energy Four-Velocity and Four-Momentum Conservation...

Table of Contents Introduction The Need for a New Transformation Einstein’s Postulates Revisited Galilean vs Lorentz Transformations Derivation of the Lorentz Transformations Time Dilation from Lorentz Transformations Length Contraction from Lorentz...

Table of Contents Introduction Historical Background and Motivation Einstein’s Postulates Galilean vs Lorentz Transformations Time Dilation Length Contraction Relativity of Simultaneity Lorentz Transformation Derivation Velocity Addition in Special Relativity Mass-Energy Equivalence Four-Vectors and Minkowski Spacetime Causality...

Table of Contents Introduction What Are Vibrations? The Simple Harmonic Oscillator Damped Oscillations Driven Oscillations and Resonance Coupled Oscillators What Are Waves? Mechanical Waves: Transverse and Longitudinal The Wave Equation Wave Parameters: Wavelength, Frequency,...