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Preface
Meditations
Prologue
Volume 0 — Mathematical & Computational Foundations
0.1 Floating-Point Arithmetic and Numerical Error
0.2 Root-Finding
0.3 Numerical Integration and Differentiation
0.4 Linear Systems and Matrix Factorizations
0.5 Eigenvalues, Diagonalization, and the SVD
0.6 The Fast Fourier Transform
0.7 Solving Ordinary Differential Equations
0.8 Fitting and Least Squares
0.9 Symbolic Computation with SymPy
0.10 Scientific Python in the Age of AI
0.11 Random Numbers and Monte Carlo Integration
0.12 Interpolation: Polynomials, Runge’s Warning, and Splines
0.13 Optimization: Golden Sections, Amoebas, and Gradient Descent
0.14 Partial Differential Equations: Stability, Explicit and Implicit
Volume I — Elementary Mechanics
1.1 Projectile Motion with Drag
1.2 The Damped, Driven Pendulum
1.3 The Double Pendulum
1.4 Kepler Orbits and the Two-Body Problem
1.5 Coupled Oscillators and Normal Modes
1.6 Symplectic vs. Naive Integrators
1.7 The Falling Chain
1.8 The Solar System: N-Body Gravitation and the Long Game
Volume II — Analytical Mechanics
2.1 Lagrangian Mechanics with SymPy
2.2 Symmetry and Conservation: Noether’s Theorem
2.3 Hamiltonian Mechanics and Phase Flow
2.4 The Central-Force Problem and Orbits
2.5 Scattering and the Rutherford Cross-Section
2.6 Rigid-Body Rotation and the Spinning Top
2.7 Small Oscillations from a General Lagrangian
2.8 The Brachistochrone and Tautochrone
2.9 Lagrange Points and the Restricted Three-Body Problem
2.10 Hamilton–Jacobi Theory and Action-Angle Variables
2.11 Nonlinear Dynamics and Chaos: When Integrability Fails
Volume III — Classical Electrodynamics
3.1 Coulomb’s Law and the Electric Field
3.2 The Electric Potential and Electrostatic Energy
3.3 Gauss’s Law and the Differential Form
3.4 Laplace’s and Poisson’s Equations
3.5 The Multipole Expansion
3.6 Magnetostatics and the Vector Potential
3.7 Electromagnetic Induction
3.8 Maxwell’s Equations and Electromagnetic Waves
3.9 Waveguides and Cavity Resonances
3.10 Radiation
3.11 RLC and AC Circuits
3.12 The Relativistic Formulation of Maxwell’s Equations
3.13 Fields in Matter: Dielectrics, Polarization, and Magnetic Materials
3.14 Wave Optics: Diffraction, Interference, and the Fourier Lens
3.15 Waves in Media: Dispersion, Absorption, and the Fresnel Relations
3.16 Anisotropic Dielectrics
3.17 Crystal Optics: Birefringence and the Wave Surface
3.18 Separation of Variables and Sturm–Liouville
Volume IV — Special Relativity
4.1 The Crisis and the Postulates
4.2 The Lorentz Transformation, Derived
4.3 Spacetime, Minkowski Diagrams, and Four-Vectors
4.4 The Paradoxes, Computed
4.5 Four-Momentum and E = mc²
4.6 Relativistic Collisions and Decays
4.7 The Relativistic Lagrangian and Motion in Fields
4.8 A Taste of Curved Spacetime
4.9 Relativistic Optics: Doppler, Aberration, and What a Camera Sees
Volume V — Classical Statistical Mechanics
5.1 Counting: Combinatorics and Microstate Enumeration
5.2 Probability: Distributions, Expectation, and the Born Rule
5.3 The Large-N Limit: Stirling, the CLT, and Sharp Macrostates
5.4 Microstates, Entropy, Temperature, and the Boltzmann Distribution
5.5 Ergodicity: Time Averages versus Ensemble Averages
5.6 The Classical Ideal Gas: Phase-Space Volume, Chemical Potential, and the Fundamental Relation
5.7 Thermodynamic Potentials, Legendre Transforms, and the Maxwell Relations
5.8 The Partition Function and the Canonical Ensemble
5.9 The Grand Canonical Ensemble, Fluctuations, and the Equivalence of Ensembles
5.10 The Ising Model: Emergence, Symmetry Breaking, and Universality
5.11 A Taste of Non-Equilibrium: Irreversibility, the Arrow of Time, and the Approach to Equilibrium
5.12 Kinetic Theory: Collisions, Mean Free Path, and Transport
5.13 Random Walks: From Coin Flips to the Diffusion Equation
5.14 Heat Engines and Thermodynamic Cycles
5.15 The van der Waals Gas: Phase Coexistence and the Critical Point
5.16 Chemical Equilibrium and the Saha Equation
5.17 Molecular Dynamics: Periodic Boundaries, Cutoffs, and Pressure
5.18 Nucleation: The Critical Droplet and the Metastable Lifetime
Volume VI — Quantum Mechanics
6.1 Complex Vector Spaces and Inner Products
6.2 Linear Operators, Hermitian and Unitary Operators, and the Spectral Theorem
6.3 Dirac Notation, Bases, and Spectral Decomposition
6.4 The Stern–Gerlach Experiment and the Birth of the Qubit
6.5 The Postulates of Quantum Mechanics
6.6 The Pauli Matrices, Incompatible Observables, and the Uncertainty Relation
6.7 Time Evolution and the Schrödinger Equation
6.8 Qubits, the Bloch Sphere, and a First Taste of Entanglement
6.9 From Vectors to Wave Functions: The Position Representation and Continuous Spectra
6.10 The Schrödinger Equation as a PDE, Solved on a Computer
6.11 Bound States in One Dimension
6.12 The Quantum Harmonic Oscillator
6.13 Scattering, Tunneling, and Wave-Packet Dynamics
6.14 The Angular-Momentum Algebra
6.15 Orbital Angular Momentum and the Spherical Harmonics
6.16 The Three-Dimensional Schrödinger Equation and Central Potentials
6.17 The Hydrogen Atom
6.18 Spin, Magnetic Moments, and the Electron in a Magnetic Field
6.19 Addition of Angular Momenta and Clebsch–Gordan Coefficients
6.20 Identical Particles, Exchange Symmetry, and the Pauli Principle
6.21 Time-Independent Perturbation Theory and Fine Structure
6.22 The Variational Method and Variational Monte Carlo
6.23 The WKB Approximation and the Semiclassical Limit
6.24 Time-Dependent Perturbation Theory and Fermi’s Golden Rule
6.25 Bell’s Inequality and the Failure of Local Realism
6.26 The Density Matrix, Mixed States, and Decoherence
6.27 Quantum Information: Gates, Circuits, Teleportation, and Algorithms
6.28 Gauge Invariance in Quantum Mechanics: The Aharonov–Bohm Effect
6.29 Scattering in Three Dimensions: Partial Waves and the Born Approximation
Volume VII — Quantum Statistical Mechanics
7.1 Complex Analysis I: Analytic Functions and the Residue Theorem
7.2 Complex Analysis II: Causality, Kramers–Kronig, Matsubara Sums, and Steepest Descent
7.3 The Statistical Toolkit: Densities of States, Polylogarithms, and the Bose and Fermi Integrals
7.4 The Thermal Density Matrix and the Quantum Canonical Ensemble
7.5 The Quantum Oscillator at Temperature: Planck’s Occupation, Freezing Out, and the Classical Limit
7.6 Molecules: Rotation, Vibration, and the Heat-Capacity Staircase
7.7 Bose–Einstein and Fermi–Dirac: The Grand Canonical Derivation
7.8 The Classical Limit and the Thermal Wavelength: The N! Derived
7.9 The Ideal Fermi Gas at T = 0: The Fermi Sea and the Stiffness of Matter
7.10 The Fermi Gas at Finite Temperature: Sommerfeld’s 0.4%, and Two Mysteries Dissolved
7.11 White Dwarfs and the Chandrasekhar Limit: Pauli versus Gravity
7.12 Electrons in a Periodic Potential: Bloch’s Theorem and the Origin of Bands
7.13 Semiconductors: Fermi–Dirac in a Gap
7.14 The Photon Gas and Planck’s Law
7.15 Einstein’s A and B Coefficients: Thermodynamics Predicts the Laser
7.16 Phonons and the Debye Model
7.17 Bose–Einstein Condensation: The Ceiling Saturates
7.18 Quantum Paramagnets: The Brillouin Function and the Refrigerator
7.19 The Transverse-Field Ising Chain: A Phase Transition at Absolute Zero
7.20 Imaginary Time and the Quantum–Classical Mapping: Temperature Is a Length
7.21 Path-Integral Monte Carlo: Coin Flips Compute Quantum Mechanics
7.22 Eigenstate Thermalization: Why Isolated Systems Forget
Coda (optional): The Many-Body Gateway
7.23 Second Quantization: The Occupation Number Becomes the State
7.24 Green’s Functions: The Propagator at Temperature
7.25 Linear Response and Kubo: How Equilibrium Answers Questions
Volume VIII — Electronic Structure and Many-Body Matter
8.1 The Many-Electron Problem
8.2 An Exact Laboratory: Two Electrons on a Grid
8.3 Hartree–Fock I: Atoms
8.4 Hartree–Fock II: The Electron Gas
8.5 Thomas–Fermi: The First Density Functional
8.6 Hohenberg–Kohn and the Constrained Search
8.7 The Kohn–Sham Construction
8.8 Exact Conditions and the Band-Gap Problem
8.9 Tight Binding: From Chain to Graphene
8.10 Plane Waves and Pseudopotentials
8.11 Real Band Structures: The Empirical Pseudopotential Method
8.12 Berry Phase, Wannier Functions, and the SSH Model
8.13 The Hubbard Model: Correlation on a Lattice
8.14 Quasiparticles, Spectral Functions, and GW
8.15 Optical Absorption and Excitons
8.16 Time-Dependent Density-Functional Theory
8.17 BCS Superconductivity
8.18 Basis Sets, and the Error They Invent
Epilogue
E.1 The Oscillator’s Biography: One System, Eight Volumes
E.2 Four Faces of the Action: One Principle, the Whole Course
E.3 Universality: Why the Details Didn’t Matter
E.4 How We Knew: The Course’s Epistemology
Afterword
Repository
Open issue
Index
