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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number .
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Gauss's Law of Universal Gravity .
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Refractive Index
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Rotational Dynamics
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Specific Heat, Latent Heat and Calorimetry
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling .
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CPT Symmetries
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Dark Matter .
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Friedmann-Robertson-Walker Equations .
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox .
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Planck Units .
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model
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Vacuum Energy .

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets .
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Capital Structure
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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Stress-Energy-Momentum Tensor .
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Tensors
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The Area Metric
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The Dirac Equation in Curved Spacetime .
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation .
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series .
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Binomial Theorem (Pascal's Triangle)
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices) .
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1 .
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Quaternions 2 .
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Regular Polygons
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Related Rates
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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Klein-Gordon and Dirac Equations .
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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Density Operators and Mixed States .
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Entangled States .
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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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Pauli Spin Matrices
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Photoelectric Effect .
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Position and Momentum States .
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Probability Current
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Schrodinger Equation for Hydrogen Atom .
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Schrodinger Wave Equation
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors .
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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Lorentz Invariance .
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Lorentz Transform .
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Lorentz Transformation of the EM Field .
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
Last modified: January 27, 2022 ✓

Klein-Gordon and Dirac Equations -------------------------------- The Schrodinger equation is: -(h2/2m)∂2ψ/∂x2 = ih∂ψ/∂t Because the time derivative is of first order and the spacial derivative is of second order, it does not treat time and space on the same footing. As a result it is not Lorentz invariant. K-G set out to resolve this by replacing the H = p2/2m term in the SE with with its relativistic version. From SR, H = E = ±√(p2c2 + m2c4) Substituting p = -ih∂/∂x and E = -ih∂/∂t gives: H = ±√{(-ih∂/∂x)2c2 + m2c4} Therefore, the TDSE becomes, ±√{(-ih∂/∂x)2c2 + m2c4}ψ = -ih∂ψ/∂t However, this is a difficult expression to work with so Klein and Gordon squared both sides to get: {(-ih∂/∂x)2c2 + m2c4)}ψ = (-ih∂ψ/∂t)2 Which simplifies to: -h2c22ψ/∂x2 + m2c4ψ = -h22ψ/∂t2 Rearranging we get, (1/c2)∂2ψ/∂t2 - ∂2ψ/∂x + m2c2ψ/h2 = 0 (□ + μ2)ψ = 0 where μ = mc/h and □ = (1/c2)∂2/∂t2 - ∂2/∂x2 The K-G equation is used to describe spin 0 bosons in Quantum Field Theory. The problem with the Klein-Gordon equation is that it allows for negative energies since E = ±√(p2 + m2). It also allows for negative probability densities. In addition, Dirac was uncomfortable with the ∂2ψ/∂t2 term because the Schrodinger equation is only dependent on the first derivative with respect to time and not the second. Instead, he sought an alternative formulation that was first order in both space and time. Consider, ψ = exp(i(kx - ωt)) ∂ψ/∂x = ikψ Therefore, ψ = (1/ik)∂ψ/∂x ... (1) Now ∂ψ/∂t = -iωψ Substituting (1) into this gives: ∂ψ/∂t = -iω(1/ik)∂ψ/∂x = (-ω/k)∂ψ/∂x Now ω/k = 2πf/2π/λ = fλ = v so: ∂ψ/∂t = -v∂ψ/∂x This is the basic wave equation. We can also write this as: -iωψ = -vikψ ∴ (ω - vk) = 0 This equation can be satisfied in the following way: +ω = (+v)(+k) or (-v)(-k) -ω = (+v)(-k) or (-v)(+k) Letting v = c, a plot of ω versus k looks like: ω (E = hω) | left moving | right moving \ | / -c \ | / +c \ | / \ | / \|/ ---------+--------- k (p = hk) /|\ Right moving / | \ Left moving Dirac Sea / | \ Dirac Sea / | \ / | \ For each quantum state possessing a positive energy, E, there is a corresponding state with energy -E. The problem with this is that a positive energy electron would be able to lose its energy by continuously emitting photons, a process that could continue without limit as the electron descends into lower and lower energy states. Real electrons clearly do not behave in this way. Dirac's solution to this was to hypothesize that what we think of as the "vacuum" is actually the state in which all the negative energy states are filled, and none of the positive energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy. These negative energy states are referred to as the Dirac Sea. There is a Dirac Sea for both right moving and left moving particles as shown in the plot. Dirac also pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively-charged particle. In later years this "hole' was determined to be the positron. Right moving electrons and holes have +ve momentum and energy. Left moving electrons and holes have -ve momentum and energy? The idea of left and right handed particles leads to the idea that there must be 2 types of electrons. The Massless Case ----------------- Left and right moving waves: ψR = exp(i(kx - ωt)) and ψL = exp(i(kx + ωt)) The wave equations for each are: ∂ψR/∂t = -c∂ψR/∂x and ∂ψL/∂t = c∂ψR/∂x The right and left equations can be written in matrix form as: - - - - - - | ∂ψR/∂t | = -c| 1 0 || ∂ψR/∂x | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | - - - - - - We can also write the above as: - - - - - - | -iωψR | = -cα| ikψR | where α = | 1 0 | | +iωψL | | ikψL | | 0 -1 | - - - - - - - - - - | -ω | = -cα| k | | +ω | | k | - - - - From this we can write: ω = +/-αck or ω2 = c2α2k2 ... (2) The Massive Case ---------------- Now introduce relativistic mass, E = √(p2c2 + m2c4) = √(h2k2c2 + m2c4) hω = √(h2k2 + m2) setting c = 1 ω = √(k2 + m2) setting h = 1 Therefore, ω2 = k2 + m2 ... (3) How can (2) with c2 = 1 be made to look like (3)? Try something of the form: ω = αk + βm where α and β are both matrices. So, ω2 = α2k2 + β2m2 + km(αβ + βα) To get this to match with (3) we need: α2 = 1, β2 = 1 and αβ + βα = 0 - - - - α2 = | 1 0 || 1 0 | = 1  | 0 -1 || 0 -1 | - - - - - - Try β = | 0 1 | | 1 0 | - - - - - - β2 = | 0 1 || 0 1 | = 1  | 1 0 || 1 0 | - - - - - - - - - - - - αβ + βα = | 1 0 || 0 1 | + | 0 1 || 1 0 | | 0 -1 || 1 0 | | 1 0 || 0 -1 | - - - - - - - - - - - - = | 0 1 | + | 0 -1 | |-1 0 | | 1 0 | - - - - - - - - = | 0 1 | - | 0 1 | |-1 0 | |-1 0 | - - - - = 0 This is just the statement that σ3σ1 + σ1σ3 = 0 The correspondence between (2) and (3) is: ω = αk corresponds to: - - - - - - | ∂ψR/∂t | = -| 1 0 || ∂ψR/∂x | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | - - - - - - Then, ω = αk + βm corresponds to: - - - - - - - - - - i| ∂ψR/∂t | = -i| 1 0 || ∂ψR/∂x | + m| 0 1 || ψR | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | | 1 0 || ψL | - - - - - - - - - - The factor of i is required to make the equation work, i.e. i(-iωψR) = -i(iαkψR) + βmψR ∴ ω = αk + βm The latter results in the following coupled equations: i∂ψR/∂t = -i∂ψR/∂x + mψL i∂ψL/∂t = +i∂ψL/∂x + mψR What happens when the particle is at rest? Under this condition αk = 0 and ω = βm This leads to: . iψR = mψL and, . iψL = mψR How do we decouple these equations? Consider the linear combinations. Adding: . . i(ψR + ψL) = m(ψR + ψL) Which can be written as: . iψ+ = mψ+ i(-iωψ+) = mψ+ ∴ ω = m Subtract bottom from top: . . i(ψR - ψL) = -m(ψR - ψL) Which can be written as: . iψ- = -mψ- i(iωψ-) = mψ- ∴ ω = -m ψ+ is linear field operator that describes a particle with +ve energy at rest and ψ- is a linear field operator that describes a particle with -ve energy at rest. The -ve energy particles go into the Dirac Sea leaving just the +ve energy particles. ψ+ = (1/√2)(ψR + ψL) represents a linear coherent superposition of quantum states that creates a single particle with a probability of 1/2 of being left-moving or right moving. In other words, it is at rest. Without the mass term, ψ+ would create a linear superposition of a particle that is actually moving to the left and one that is actually moving to the right. The mass effectively mixes the moving ψR and ψL waves to create a wave that is not moving at all. Extension to 3 Dimensions ------------------------- Now, from above we had: ω2 = k2 + m2 = kx2 + ky2 + kz2 + m2 ... (4) Also, from above we had: ω = αk + βm Which becomes: ω = αxkx + αyky + αzkz + βm This leads to, ω2 = αx2kx2 + αy2ky2 + αz2kz2 + β2m2 + 12 other terms that equal 0 ... (5) In order to match equations (4) and (5) we must have: αx2 = αy2 = αz2 = β2 = 1 αiαj + αjαi = 2δij αiβ + αjβ = 0 After adding back c and h we get the complete Dirac Equation as written by Dirac: ih∂ψ/∂t = -ihi∂ψ/∂x + βmc2ψ Which can be rewritten in terms of the momentum operator, p, as: ih∂ψ/∂t = (cαip + βmc2)ψ Covariant Form -------------- Restate the Dirac equation with h = c = 1. i∂ψ/∂t + iαi∂ψ/∂x - βmψ = 0 Now define γ0 = β and multiply everything by γ0. Therefore, iγ0∂ψ/∂t + iγ0αi∂ψ/∂x - (γ0)2mψ = 0 Now (γ0)2 = 1. Therefore, iγ0∂ψ/∂t + iγ0αi∂ψ/∂x - mψ = 0 Now define γi = γ0αi. Therefore, iγ0∂ψ/∂t + iγi∂ψ/∂x - mψ = 0 This can be written as: iγμμψ - mψ = 0 Or, (iγμμ - m)ψ = 0 This is the covariant form. Noting that p -> i∂/∂x and p0 -> ∂/∂t we can write: (iγμpμ - m)ψ = 0 This is the momentum form. The Gamma Matrices ------------------ Dirac realized that the solution to this equation involved 4 x 4 matrices. These became known as the GAMMA MATRICES. - -   | 1 0 : 0 0 |   | 0 1 : 0 0 | - - γ0 = | .....:...... | = | I 0 |   | 0 0 : -1 0 | | 0 -I |   | 0 0 : 0 -1 | - - - - - -   | 0 0 : 0 1 |   | 0 0 : 1 0 | - - γ1 = | ......:...... | = | 0 σx |   | 0 -1 : 0 0 | | -σx 0 |   | -1 0 : 0 0 | - - - - - -   | 0 0 : 0 -i |   | 0 0 : i 0 | - - γ2 = | ......:...... | = | 0 σy |   | 0 i : 0 0 | | -σy 0 |   | -i 0 : 0 0 | - - - - - -   | 0 0 : 1 0 |   | 0 0 : 0 -1 | - - γ3 = | ......:...... | = | 0 σz |   | -1 0 : 0 0 | | -σz 0 |   | 0 1 : 0 0 | - - - - The σ matrices are the PAULI SPIN matrices. The algebraic structure represented by the γ matrices had been created some 50 years earlier than Dirac by the English mathematician W. K. Clifford. In fact, (γo)2 = 1, (γ1)2 = (γ2)2 = (γ3)2 = -1 form the basis of a CLIFFORD ALGEBRA. However, Dirac was unaware of this previous work. The Clifford algebra has the defining relationship: {γμν} = γμγν + γνγμ = 2ημνI It is important to realise that the Dirac Gamma matrices are not four-vectors. They are constant matrices which remain invariant under a Lorentz transformation. Since the γ's; are 4 x 4 matrices the solutions to the Dirac equation have to be 4-component entities. These entities are referred to as the DIRAC SPINORS. Computation of these spinors is easier if we note that the γ matrices can be written in terms of 2 component I and σ matrices as: - - - - γi = | 0 σi | where i = x, y, z and γ0 = | I 0 |  | -σi 0 |   | 0 -I | - - - - Likewise we can decompose ψ into 2 two-component Lorentz spinors φ and χ as: - - ψ = | φ | | χ | - - This decomposition reflects the fact that the Dirac spinors form a reducible representation consisting of irreducible representations, φ and χ. These are the representations used in the quantum mechanics of spin. Now, using the covariant form (iγμμ - m)ψ = 0 where c and h have been set to 1 we get: - - iγ0∂/∂t = | E 0 |   | 0 -E | - - - - iγi∂/∂xi = | 0 σ.p |    | -σ.p 0 | - - Where, - - σ.p = σxpx + σypy + σzpz = | pz (px - ipy)|        |(px + ipy) -pz | - - Returning to the covariant form: (iγ0∂/∂t + iγi∂/∂x - m)ψ = 0 ∴ iγ0∂ψ/∂t = -iγi∂ψ/∂xi + mψ ∴ γ0E = -γiσ.p + mψ - - - - - - - - γ0E| φ | = -| 0 σ.p || φ | + m| φ |   | χ | | -σ.p 0 || χ | | χ | - - - - - - - - This gives: Eφ = -σ.pχ + mφ -Eχ = σ.pφ + mχ ∴ φ = (-σ.p/(E - m))χ ≡ (σ.p/(-E + m))χ and, χ = (σ.p/(-E - m))φ ≡ -(σ.p/(E + m))φ We can also solve this in terms of the c characteristic quation, (E - mI)ψ = 0: - - - - - - | E σ.p || φ | = m| φ | | -σ.p -E || χ | | χ | - - - - - - - - - - | E - m σ.p || φ | = 0 | -σ.p -E - m || χ | - - - - (E - m)φ + σ.pχ = 0 -σ.pφ + (-E - m) = 0 Which is the same as before. We can write these 2 equations as: 1. 1φ + σ.p/(E - m)χ = 0 2. -σ.pφ/(-E - m) + 1χ = 0 Combining gives: - - - - uφ = | 1 |φ uχ = | σ.p/(E - m) |χ   | -σ.p/(-E - m) |  | 1 | - - - - If we let: - - - - - - - - φ = | 1 | : | 0 | and χ = | 1 | : | 0 | | 0 | | 1 | | 0 | | 1 | - - - - - - - - We get the 4-component Dirac spinors, ui: - - - -   | 1    |   | 0    | u1 = N| 0    | u2 = N| 1    |   | pz/(E + m)  |   | (px - ipy)/(E + m) |   | (px + ipy)/(E + m) |   | -pz/(E + m)   | - - - - - - - -   | -pz/(-E + m)   |   | -(px - ipy)/(-E + m) | u3 = N| -(px + ipy)/(-E + m) | u4 = N| pz/(-E + m)   |   | 1    |   | 0    |   | 0    |   | 1    | - - - - Where N is a normalization factor. Mathematically, these spinor objects describe quantum mechanical spin states. Spinors are different from vectors because they behave differently under rotations. The above describes 4 different spin states. There are 2 spin states for E = +√(p2c2 + m2c4) and 2 states for E = -√(p2c2 + m2c4) . For a FREE fermion the wavefunction is the product of a plane wave and a spinor, u(p) or v(p). Thus: spin up particle: ψ = u1(p)exp(-ipx)exp(-iEt) spin down particle: ψ = u2(p)exp(-ipx)exp(-iEt) spin up antiparticle: ψ = u3(-p)exp(-ipx)exp(-iEt) spin down antiparticle: ψ = u4(-p)exp(-ipx)exp(-iE) Note: exp(-iωt) ≡ exp(-iEt/h) ≡ exp(-imc2t/h) For a particle at rest: Eγ0ψ = mψ - - - - - - | E 0 0 0 || u1 | | u1 | | 0 E 0 0 || u2 | = m| u2 | | 0 0 -E 0 || u3 | | u3 | | 0 0 0 -E || u4 | | u4 | - - - - - - Using the characteristic polynomial gives: - - - - | E - m 0 0 0 || u1 | | 0 E - m 0 0 || u2 | = 0 | 0 0 -E - m 0 || u3 | | 0 0 0 -E - m || u4 | - - - - It is a property that the eigenspinors of a diagonal matrix are:   - -  - -  - -   - -   | 1 |   | 0 |  | 0 |  | 0 | u1 = | 0 | u2 = | 1 | v1 = | 0 | v2 = | 0 |   | 0 |  | 0 |  | 1 |  | 0 |   | 0 |  | 0 |  | 0 |  | 1 |   - -  - -  - -  - - and the eigenvalues are m, m, -m, -m. This is same result as before when p = 0; In this case, ψi = uiexp(-imc2t/h) which represents the time evolution of ψi. The Klein-Gordon Equation Revisited ----------------------------------- Consider: (iγμpμ - m)(iγνpν + m)ψ = 0 This equals: (-γμγνpμpν - m2)ψ = 0 γμγν is rather complicated and consists of terms involving (γ0)2p02 + (γ1)2p12 + ... + (γ0γ1 + γ1γ0) + ... - m2. Now, from the Clifford algebra, {γμν} = 2ημνI = 0 if μ ≠ ν Knowing this we are left with: (-γμγνpμpν - m2)ψ = [(γ0)2p02 + (γ1)2p12 + (γ2)2p22 + (γ3)2p32 - m2]ψ = 0 Now, (γo)2 = 1, (γ1)2 = (γ2)2 = (γ3)2 = -1 So we are left with: [p02 - p12 - p22 - p32 - m2]ψ = 0 Or, [ημνpμpν - m2]ψ = 0 where ημν = (+, -, -, -) Or, [-ημνpμpν + m2]ψ = 0 Or, [pμpμ - m2]ψ = 0 Or, [-pμpμ + m2]ψ = 0 Note: pμ = (E/c,p1,p2,p3) pμ = (-E/c,p1,p2,p3) pμpμ = -E2/c2 + |p|2 = (-m2c4 + p2c2)/c2 = -m2c2 + p2 This is the Klein-Gordon equation.