Quantum 101

Table of Contents Introduction Why Simulate Classical Systems? Newtonian Mechanics and Time Integration Lagrangian and Hamiltonian Formulations Numerical Methods for Classical Simulations Simulating Rigid Body Dynamics Molecular Dynamics (MD) N-Body Simulations in...

Table of Contents Introduction Why Use Numerical Methods in Physics? Types of Numerical Problems Root-Finding Algorithms Numerical Differentiation and Integration Solving Ordinary Differential Equations (ODEs) Solving Partial Differential Equations (PDEs) Finite Difference...

Table of Contents Introduction Why Computational Tools Matter in Physics Categories of Computational Methods Symbolic Computation Tools Numerical Libraries and Environments Data Visualization and Analysis Programming Languages Commonly Used Monte Carlo Simulations Differential...

Table of Contents Introduction What Does Mathematical Rigor Mean? The Role of Rigor in Theoretical Physics Historical Evolution: From Intuition to Formalism Common Tools Requiring Rigor: Limits, Derivatives, and...

Table of Contents Introduction What Is Set Theory? Basic Set Operations Subsets, Power Sets, and Cartesian Products Relations and Functions Types of Sets: Finite, Infinite, Countable, and Uncountable Russell’s Paradox and...

Table of Contents Introduction What Is Topology? Topological Spaces and Open Sets Basis and Subbasis for a Topology Closed Sets and Limit Points Continuity in Topological Terms Homeomorphisms and Topological Invariants Compactness:...

Table of Contents Introduction What Is Functional Analysis? Normed and Banach Spaces Inner Product and Hilbert Spaces Linear Operators and Functionals Bounded and Unbounded Operators Dual Spaces and the Hahn–Banach Theorem The...

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...