Quantum 101

Table of Contents Introduction What Is a Tensor? Scalars, Vectors, and Higher-Rank Tensors Tensor Notation and Index Conventions Contravariant and Covariant Tensors The Metric Tensor and Index Raising/Lowering Tensor Transformation Laws Symmetry...

Table of Contents Introduction Overview of SU(2), SO(3), and U(1) U(1): The Simplest Lie Group U(1) in Physics: Electromagnetism and Phase Invariance SO(3): The Rotation Group in Three Dimensions Properties...

Table of Contents Introduction What Are Lie Groups? Examples of Lie Groups Smooth Manifolds and Group Structure What Is a Lie Algebra? The Lie Bracket and Commutators Matrix Lie Groups and...

Table of Contents Introduction What Is Group Theory? Definitions: Groups, Subgroups, and Cosets Homomorphisms and Isomorphisms Group Actions and Orbits Symmetry in Physics Permutation Groups and Parity Lie Groups and Lie Algebras Representations...

Table of Contents Introduction What Is a Stochastic Process? Index Sets and State Spaces Types of Stochastic Processes Stationarity and Ergodicity Markov Processes Discrete-Time Markov Chains Continuous-Time Markov Processes Poisson Processes Birth-Death Processes Brownian Motion...

Table of Contents Introduction What Is a Statistical Distribution? Types of Distributions: Discrete vs Continuous Probability Mass Functions (PMFs) Probability Density Functions (PDFs) Cumulative Distribution Function (CDF) Bernoulli and Binomial Distributions Poisson...

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...