Quantum 101

Table of Contents Introduction What Is Probability? Sample Spaces and Events Axioms of Probability Conditional Probability Bayes’ Theorem Independent and Dependent Events Random Variables: Discrete and Continuous Probability Distributions Expectation and Variance Common Discrete Distributions Common...

Table of Contents Introduction What Are Partial Differential Equations? Order, Linearity, and Classification First-Order PDEs Method of Characteristics Second-Order Linear PDEs Classification: Elliptic, Parabolic, Hyperbolic Canonical Forms and Transformation of Variables The Heat...

Table of Contents Introduction What Is an Ordinary Differential Equation? Order and Linearity First-Order ODEs Separable Equations Linear First-Order Equations Exact Equations and Integrating Factors Second-Order Linear ODEs Homogeneous and Inhomogeneous Equations Characteristic Equation...

Table of Contents Introduction Motivation: Beyond Classical Functions The Dirac Delta Function Properties of the Delta Function Representations and Approximations Delta Function in Multiple Dimensions Derivatives of the Delta Function Heaviside Step...

Table of Contents Introduction Motivation: Why Fourier Analysis? Fourier Series: Periodic Function Decomposition Orthogonality of Trigonometric Functions Fourier Coefficients Convergence of Fourier Series Even and Odd Functions Complex Exponential Form of Fourier...

Table of Contents Introduction What Are Complex Numbers? Algebra of Complex Numbers Complex Plane and Polar Representation Euler’s Formula and Roots of Unity Complex Functions: Definition and Examples Analytic Functions and...

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...