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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 Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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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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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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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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Wien's Displacement Law
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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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Geometries of the Universe
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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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Penrose Diagrams
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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

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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Dividend Discount Formula
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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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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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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 Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

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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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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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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Double Angle Formulas
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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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Sets, Groups, Modules, Rings and Vector Spaces
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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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Symmetric Groups
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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 Integers Modulo n Under + and x
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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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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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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 .
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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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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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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Basic Relationships
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Bell's Theorem
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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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Entangled States
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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 Exclusion Principle
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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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Schrodinger Wave Equation (continued)
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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 Observer Effect
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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 4 - Potential
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Energy and Momentum, E = mc2
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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 Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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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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BCS Theory
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Introduction to Superconductors
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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
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Formulas
Last modified: January 26, 2018

Klein-Gordon and Dirac Equations -------------------------------- It can be shown that the Schrodinger equation 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 equivalent. From SR, H = E = ±√(p2c2 + m2c4) substitute p for -ih∇ and E = -ih∂/∂t H = ±√{(-ih∇)2c2 + m2c4} Therefore, the TDSE becomes, ±√{(-ih∇)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∇)2c2 + m2c4)}ψ = (-ih∂ψ/∂t)2 which simplifies to -h2c22ψ + m2c4ψ = -h22ψ/∂t2 Rearranging we get, (1/c2)∂2ψ/∂t2 - ∇2ψ + m2c2ψ/h2 = 0 (□ + μ2)ψ = 0 where μ = mc/h and □ = (1/c2)∂2/∂t2 - ∇2 ... the d'Alembert operator 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 (E = ±√(p2 + m2)) and negative probability densities. Also, Dirac was uncomfortable with the ∂2ψ/∂t2 term because the Schrodinger equation only dependent on the first derivative with respect to time not the second He sought an alternative formulation that was first order in both space and time. Consider, ψ = exp(i(kx - ωt)) ∂ψ/∂t = -iωψ ∂ψ/∂x = ikψ Therefore, ψ = (1/ik)∂ψ/∂x Substituting back into ∂ψ/∂t = -iωψ gives, ∂ψ/∂t = -iω(1/ik)∂ψ/∂x = (-ω/k)∂ψ/∂x = -v∂ψ/∂x since ω = 2πf and k = 2π/λ and v = fλ This is the basic wave equation. We can also write this (from above) as: -iω = -vik ∴ (ω - vk) = 0 This equation can be satisfied with both positve and negative values of ω and k. ∴ ω = ±vk A plot of ω versus k looks like: ω (E) | left moving | right moving \ | / -v \ | / +v \ | / \ | / \|/ ---------+--------- k (p) /|\ 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 is that a positive energy electron would be able to shed 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. Left and right moving waves: ψR = exp(i(kx - ωt)) and ψL = exp(i(kx + ωt)) The wave equations for each are: i∂ψR/∂t = -iv∂ψR/∂x and i∂ψL/∂t = iv∂ψR/∂x where we have multiplied both sides by i. The right and left equations can be written in matrix form as: - - - - - - i| ∂ψR/∂t | = -iv| 1 0 || ∂ψR/∂x | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | - - - - - - Also, ∂ψR/∂t = -iωψR and ∂ψR/∂x = +ikψR and ∂ψL/∂t = +iωψL and ∂ψL/∂x = +ikψL Thus, we can write the above matrix equation as: - - - - - - i| -iωψR | = -ivα| ikψR | where α = | 1 0 | | +iωψL | | ikψL | | 0 -1 | - - - - - - From this we can write: ω = +/-αk or ω2 = α2k2 ... A. 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 ... B. How can A be made to look like B? 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 B 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 || 1 0 | + | 0 1 || 1 0 | | 0 -1 || 0 -1 | | 1 0 || 0 -1 | - - - - - - - - - - - - = | 0 1 | + | 0 -1 | |-1 0 | | 1 0 | - - - - - - - - = | 0 1 | - | 0 1 | |-1 0 | |-1 0 | - - - - = 0 The correspondence between A and B is, - - - - - - ω = αk corresponds to i| ∂ψR/∂t | = -iv| 1 0 || ∂ψR/∂x | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | - - - - - - Then, - - - - - - - - - - ω = αk + βm corresponds to i| ∂ψR/∂t | = -iv| 1 0 || ∂ψR/∂x | + m| 0 1 || ψR | | ∂ψL/∂t | | 0 -1 || ∂ψL/∂x | | 1 0 || ψL | - - - - - - - - - - Which results in the following coupled equations, i∂ψR/∂t = -iv∂ψR/∂x + mψL i∂ψL/∂t = +iv∂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 linear combinations. Adding: . iψ+ = mψ+ Subtract bottom from top: . iψ- = -mψ- ψR is linear field operator that describes a particle with +ve energy at rest and ψL 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+ = ψR + ψL creates a particle with a probability of 1/2 of being a left-moving or right moving. Thus, ψ+ is linear coherent superposition of quantum states that creates a single particle at rest. Extension to 3 Dimensions ------------------------- Now, from above we had: ω2 = k2 + m2 = kx2 + ky2 + kz2 + m2 ... C. 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 ... D. In order to match equations C and D we must have, αx2 = αy2 = αz2 = β2 = 1 After adding back constants and making v = c, 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 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. The γ's can be written as, - - - - γi = | 0 σi | where i = x, y, z and γ0 = | I 0 |  | -σi 0 |   | 0 -I | - - - - Now, - - - - γi.p = | 0 σ.p | and γ0m = | mI 0 |   | -σ.p 0 |  | 0 -mI | - - - - Thus, - - (γi.p + βm) = | mI σ.p |   | -σ.p -mI | - - Rembering that E = i∂/∂t. we can write the eigenvalue equation as: - - - -  - - E| φ | = | mI σ.p || φ | | χ | | -σ.p -mI || χ | - - - -  - - Where φ and χ are the 2 component left and right handed spinors. These are just 2 coupled equations: Eφ = mφ + σ.pχ ∴ φ = (σ.p/(E - m))χ ≡ (-σ.p/(-E + m))χ Eχ = -σ.pφ - mχ ∴ χ = (-σ.p/(E + m))φ Where, - - σ.p = σxpx + σypy + σzpz = | pz (px - ipy)|        |(px + ipy) -pz | - - Since the γ's; are 4 x 4 matrices the solutions to the Dirac equation has to be a 4-component quantity. This quantity is referred to as the DIRAC SPINOR. The eigenvalue equation yields 4 eigenspinors: - - - -   | 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) | v1 = N| (-px - ipy)/(-E + m) | v2 = N| pz/(-E + m)   |   | 1    |   | 0    |   | 0    |   | 1    | - - - - Where E = +/-√(p2c2 + m2c4) and 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. Two spin states for E = +√(p2 + m2) and 2 states for E = -√(p2 + m2). For a FREE fermion the wavefunction is the product of a plane wave and a spinor, u(p)/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) = v1(p)exp(ipx)exp(iEt) spin down antiparticle: ψ = u4(-p)exp(-ipx)exp(-iE) = v1(p)exp(ipx)exp(iEt) Note: Some notations use ψ = u(p)exp(-ipμxμ) where pμ = (E, -px, -py, -pz) and xμ = (t, x, y, z) so that -ipμxμ = -i(Et - x.p) Note: exp(-iωt) ≡ exp(-iEt/h) ≡ exp(-im2ct) Note: For stationary particles p = 0 and the eigenvalue equation becomes: - - - -  - - E| φ | = | mI 0 || φ | | χ | | 0 -mI || χ | - - - -  - - This is equivalent to: Eψ = γ0mψ Or, since (γ0)2 = 1, Eγ0ψ = mψ Now, ψ = u(p)exp(-iEt) Therefore, Eγ0u(p) - mu(p) = 0 Or, - - | 1 0 0 0 | E| 0 1 0 0 |u(p) = mu(p) | 0 0 -1 0 | | 0 0 0 -1 | - - The eigenspinors become:   - -  - -  - -   - -   | 1 |   | 0 |  | 0 |  | 0 | u1 = | 0 | u2 = | 1 | v1 = | 0 | v2 = | 0 |   | 0 |  | 0 |  | 1 |  | 0 |   | 0 |  | 0 |  | 0 |  | 1 |   - -  - -  - -  - - with eigenvalues m, m, -m, -m This describes 4 different spin states. Two spin states for E = +m and 2 states for E = -m. The Klein-Gordon Equation Again ------------------------------- Consider: (iγΚpΚ - m)(iγλpλ + m)ψ = 0 This equals: (-γΚγλpΚpλ - m2)ψ = 0 γΚγλ is rather complicated and consists of terms involving (γ0)2p02 + (γ1)2p12 + ... + (γ0γ1 + γ1γ0) + ... - m2. Now, from the Clifford algebra, {γΚλ} = 2ηΚλI = 0 if Κ ≠ λ Knowing this we are left with: (-γΚγλpΚpλ - 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, [ηΚλpΚpλ - m2]ψ = 0 where ηΚλ = (+, -, -, -) Or, [-ηΚλpΚpλ + m2]ψ = 0 Or, [pμpμ - m2]ψ = 0 Or, [-pμpμ + m2]ψ = 0 This is the Klein-Gordon equation.