Quantum 101

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...

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...