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Astronomy

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Celestial Coordinates
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Distance Units
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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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Elastic and Inelastic Collisions
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Rotational Dynamics
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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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Penrose Diagrams
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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Geometries of the Universe
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Inflation Theory
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Stephen Hawking's PhD Thesis
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Finance and Accounting

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Brownian Model of Financial Markets
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Lecture Notes on International Financial Management
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Quantum Gravity
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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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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group 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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Contravariant and Covariant Components of a Vector
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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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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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Matrices and Determinants
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Orthogonal Curvilinear Coordinates
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Quaternions 1
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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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Confidence Intervals
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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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 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 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 Qubit
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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

test

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test

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: May 18, 2020

The Essential Mathematics of Quantum Mechanics ---------------------------------------------- Linear Algebra Representation (Dirac Notation) ---------------------------------------------- In classical mechanics, all possible states of a system are represented by a phase space. In QM a state is a complex function that lives in an abstract complex vector space called HILBERT SPACE. In other words, vectors are represented by complex functions and should not be viewed as vectors in the traditional sense (i.e. velocity, acceleration etc.). The rules for functions are defined in exactly the same way but since there are infinitely many x values the summation sign is replaced by the integral over all values of x. Before we go any further we need to introduce the DIRAC NOTATION. Dirac Notation -------------- We denote BRAs and KETs as <φ| and |ψ> respectively. They can be interpreted in several different ways: 1. |ψ> is a vector in Hilbert space. 2. <ψ| is a dual vector in conjugate Hilbert space. 3. <φ|ψ> is the inner product ψ with φ. In integral form this: <φ|ψ> = ∫φ*(x)ψ(x)dx is the degree to which ψ is the same as φ. This is called the OVERLAP INTEGRAL. <ψ|ψ> = 1 4. <ψ|φ> = <φ|ψ>* 5. <φ|O|ψ> = <φ|[O|ψ>] = [<φ|O]|ψ>. In integral form this is: <φ|O|ψ> = ∫φ*(x)Oψ(x)dx 6. OM|ψ> = O|[M|ψ>]> Fourier Series, Basis Vectors and Completeness ---------------------------------------------- Vector spaces have bases. We denote a basis by ei. If the basis is orthonormal (orthogonal and unit length) then: <ei|ej> = δij We can expand a vector, V, as: V = ΣViei i The equivalent in QM is the FOURIER SERIES expansion given by: |ψ> = Σαi|ei> i Where ψ is the WAVEFUNCTION and αi are coefficients. The above is referred to as COMPLETENESS. It means that any vector from the space under consideration can be represented as linear combination of basis vectors from this space. The components of V can be found by projecting the vector onto the respective directions. This is done by taking the dot product of V with the corresponding basis vector. ej.V = (ΣieiVi).ej = Σi(ei.ej)Vi = Vj since ei.ej = δij Likewise, to get αi we proceed as follows: <ej|ψ> = Σαi<ej|ei> i = αj In integral form of this is: αj = ∫ej*ψ dx So, basically, αm measures the amount of ψ(x) in the corresponding basis vector. Dyadic (Outer) Product ---------------------- From before we had: |ψ> = Σαi|ei> ≡ Σ|eii Now αi = <ei|ψ>. Therefore, |ψ> = Σ|ei><ei|ψ> i So the implication is that |ei><ei| = 1 Or, in general: |ei><ej| = δij This is an important formula in QM and is called a DYAD. To see the difference between the dyadic product and the inner product, consider two regular vectors. The inner product is given by: - - - - | 1 0 0 || 1 | = 1 - - | 0 | | 0 | - - Now consider the Dyadic Product: - - - - - - | 1 || 1 0 0 | = | 1 0 0 | | 0 | - - | 0 0 0 | | 0 | | 0 0 0 | - - - - Or, in general: - -    | 1 0 0 | |ei><ej| = | 0 1 0 | = I    | 0 0 1 | - - This means that we can write: |ψ> = Σ|ei><ei|ψ> i Likewise, <ψ| = Σ<ψ|ei><ei| i This will be very useful as we move forward. Linear Operators - The Matrix Element ------------------------------------- Simplistically, a linear operator, O, can be regarded as a machine that takes a vector as an input and outputs another vector i.e. |O> = Σαi|ei> <ej|O> = Σαi<ej|ei> = Σαiδji i = αi |O> = Σei><ei|O> Now consider: O|ψ> = |ψ'> <ei|O|ψ> = <ei|ψ'> = βi Σ<ei|O|ej><ej|ψ> = Σ<ei|O|ejj j  j = βi <ei|O|ej> = Oij is called the MATRIX ELEMENT. We can write this as: - - - - - - | O11 O12 O13 ... || α1 | | β1 | | O21 O22 O23 ... || α2 | = | β2 | | O31 O32 O33 ... || α3 | | β3 | | .   .   .  ... || .  | | .  | - - - - - - Or, equivalently: - - | <e1|O|e1> <e1|O|e2> <e1|O|e3> ... | | <e2|O|e1> <e2|O|e2> <e2|O|e3> ... | | <e3|O|e1> <e3|O|e2> <e3|O|e3> ... | | . . .      ... | - - If the ei's are orthogonal then: - - | <e1|O|e1> 0 0 ....... | | 0 <e2|O|e2> 0 ....... | | 0 0 <e3|O|e3> ... | | . . .   ... | - - The integral form of Oij is: Oij = ∫ei*Oej dx Hermitian Operators ------------------- <ei|O|ej> = <ei|ej'> <ei|O|ej> = <ei'|ej> Now <ei|ej'>* = <ej'|ei>. Therefore, <ei|O|ej>* = <ej|O|ei> Oij* = 0ji This referred to as HERMITIAM CONJUGATION and O is an HERMITIAN OPERATOR.. Hermitian Matrices ------------------ Hermitian matrices as operators are the analog of 'real' for numbers. For a number to be real N* = N. For an operator this implies that O = O. This requires that the diagonal elements must be real (incuding 0) since Oii must equal Oii*. They can have complex elements but elements reflected about the diagonal must be complex conjugates of each other. And, for obvious reasons, they must be square. If the matrix has complex elements it cannot be SYMMETRIC (i.e. O = OT). Hence, an Hermitian matrix that has only real elements is forced to be symmetric. For example, - - O = | 0 -i | is Hermitian but not symmetric | i 0 | (O = OT). - - - - - - O = | 0 1 | and | 1 0 | are both Hermitian | 1 0 | | 0 -1 | AND symmetric. - - - - It can easily be seen that these matrices satisfy: O = O Hermitian operators correspond to the observables of a system. Unitary Operators ----------------- A unitary operator preserves the lengths and angles between vectors, and it can be considered as a type of rotation operator in abstract vector space. Like Hermitian operators, the eigenvectors of a unitary matrix are orthogonal. However, its eigenvalues are not necessarily real. Consider: U|ψ> = |ψ'> <φ'| = <φ|U <φ'|ψ'> = <φ|UU|ψ> For <φ'|ψ'> to equal <φ|ψ> requires that UU = I Example: - - O = | 0 -i | is unitary. | i 0 | - - Proof: - - - - U = | 0 -i | U = | 0 -i | | i 0 |   | i 0 | - - - - - - - - - - | 0 -i || 1 | = | 0 | | i 0 || 0 | | i | - - - - - - - - - - - - | 1 0 || 0 -i | = | 0 -i | - - | i 0 | - - - - - - - - - - - - <| 1 0 ||| 0 -i || 0 -i ||| 1 |> - - | i 0 || i 0 | | 0 | - - - - - - - - - - | 1 0 || 1 | = 1 - - | 0 | - - - - - - | 0 -i || 0 | = 1 - - | i | - - The Eigenvalue Equation ----------------------- O|ψ> = λψ|ψ> O is Hermitian ψ = eigenvectors λI = eigenvalues The eigenvalue equation can also be written in terms of the matrix elements as: O|ψ> = E|ψ> OΣ|ei><ei|ψ> = E|ψ> ΣO|ei><ei|ψ> = E|ψ> Multiply by <ej| Σ<ej|O|ei><ei|ψ> = E|<ej|ψ> ΣOij<ei|ψ> = E|<ej|ψ> Explicitly, this looks like: - - - - - - | O11 O12 O13 ... || α1 | | β1 | | O21 O22 O23 ... || α2 | = λ| β2 | | O31 O32 O33 ... || α3 | | β3 | | .   .   .  ... || .  | | .  | - - - - - - Eigenvalues of Hermitian matrices are real. Proof: <I|O| = <I|λI* <I|O| = <I|λI* <I|O|I> = λI<I|I> <I|O|I> = <I|I>λI* λI* = λI The eigenvectors of symmetric Hermitian matrices (ones with only real elements) are always orthogonal as long as the eigenvectors have different eigenvalues. Proof: O|I> = λI|I> <J|O = λJ<J| <J|O|I> = λI<J|I> <J|O|I> = λJ<J|I> (λI - λJ)<J|I> = 0 J and I must be orthogonal if (λI - λJ) ≠ 0 If 2 eigenvectors have the same eigenvalue they are said to be DEGENERATE. However, in this case it is still possible to find a set of ORTHONORMAL eigenvectors by using the GRAM SCHMIDT process: wa = va - (va.vb/vb.vb)vb Where va and vb are non orthogonal vectors and wa is the orthonormalized version of va. Lets look an example. Consider, - - | 0 1 1 | | 1 0 1 | | 1 1 0 | - - This has the following eigenvalues and eigenvectors. - -    | 1 | λ1 = 2 v1 = | 1 |    | 1 | - - - - - -    | -1 |   | -1 | λ2 = -1 v2 = | 1 | v3 = | 0 |    | 0 |   | 1 | - - - - v1 is orthogonal to v2 and v3 but v2 is not orthogonal to v3. We can use Gram Schmidt to replace v3 with w3. w3 = v3 - (v3.v2/v2.v2)v2 - - - - - -   | -1 | | -1 | | -1/2 | w3 = | 0 | - (1/2)| 1 | = | -1/2 |   | 1 | | 0 | | 1 | - - - - - - In order for this to work we need to first normalize the 3 vectors. |v1| = √[12 + 12 + 12] = √3 |b2| = √[12 + 02 + (-1)2] = √2 |w3| = √[(1/2)2 + (-1)2 + (1/2)2] = √(3/2) Therefore: - -    | 1/√3 | λ1 = 2 v1 = | 1/√3 |    | 1/√3 | - - - -    | -1/√2 | λ1 = -1 v2 = | 1/√2 |    | 0 | - - - -    | -1/√6 | λ1 = -1 w3 = | -1/√6 |    | √6/3 | - - These 3 vectors are now orthonormal to each other. Eigendecomposition (Diagonalization) ------------------------------------ A matrix is diagonalizable if there exists a unitary matrix, U, that satisfies the relationship: U-1OU = Λ The following procedure is used to diagonalize a matrix. 1. Find the eigenvalues and eigenvectors of O. If the eigenvectors are linearly independent proceed to step 2 otherwise the matrix is not diagonizable. 2. Write the eigenvectors, vn as column vectors in a matrix. - - U = | vn vn ... | -^ ^ - | | ev1 ev2 3. Compute U-1OU = Λ. The result will be a diagonal matrix of the eigenvalues. - - | λ1 0 .... 0 | λ = | 0 λ2 ... 0 | | : : : | | 0 0 ... λn | - - The set of all possible eigenvalues of Λ is called its SPECTRUM. Example: Lets return to the matrix we had above: - - | 0 1 1 | O = | 1 0 1 | | 1 1 0 | - - Without Gram Schmidt we can write U as: - - | 1 1 -1 | U = | 1 -1 0 | | 1 0 1 | - - - -    | 1/3 1/3 1/3 | U-1 = | -1/3 2/3 -1/3 |    | -1/3 -1/3 2/3 | - - U-1OU gives: - - | 2 0 0 | | 0 -1 0 | | 0 0 -1 | - - So the diagonal consists of the eigenvalues of the original matrix. If we were to repeat the process using the orthonormal basis obtained from Gram Schmidt we get: - - | 1/√3 -1/√2 -1/√6 | U = | 1/√3 1/√2 -1/√6 | | 1/√3 0 √6/3 | - - - -    | 1/√3 1/√3 1/√3 | U-1 = | -1/√2 1/√2 0 |    | -1/√6 -1/√6 √6/3 | - - Again U-1OU gives: - - | 2 0 0 | | 0 -1 0 | | 0 0 -1 | - - Therefore, diagonalizing O after going through the Gram Schmidt process yields the same result as before. So orthogonality/orthonormality is not a factor in determining whether or not the matrix can be diagonalized. The only requirement is that U-1OU = Λ. Using Gram Schmidt is computationally easier for large matrices because in an orthonormal basis U-1 = UT and UT is easier to calculate for large matrices. U-1OU = Λ can be rearranged as follows: Multiply from the left by U to get: OU = UΛ Multiply from the right by U-1 to get: O = UΛU-1 Eigenvectors as Bases --------------------- The fact that the eigenvectors of Hermitian operators are mutually orthogonal means that they form a complete basis spanning the vector space. Therefore, the vi's from the previous discussion are equivalent to the ei's. Now that we have a basis we can write any function in that as a linear combination of the eigenfunctions of the operator. Expectation (Average) Value of an Observed Quantity --------------------------------------------------- The expectation value of O is denoted by, <O> and is defined as: <O> = <ψ|O|ψ> = Σ<ψ|O|ei><ei|ψ> i But O|ei> = λi|ei> Therefore, <ψ|O|ψ> = Σλi<ψ|ei><ei|ψ> i = Σ|<ψ|ei>|2λi i = ΣPriλi i Where Pri is the probability obtaining the eigenvalue λi. The expectation value can be interpreted as the average value of the statistical distribution of eigenvalues obtained from a large number of measurements. Position and Momentum Operators ------------------------------- We now look at 2 important operators - position and momentum. Position Operator ----------------- O|ψ> = λ|ψ> Check that O is Hermitian <ψ(x)|O|ψ(x)> = ∫ψ*Oψdx = real Therefore O is Hermitian. We can rewrite the eigenvalue equation as: (x - λ)|ψ(x)> = 0 Therefore ψ(x) must look like: >ε< ||^ || 1/ε ε is very small ||v ----------------- ------------------- x The eigenvectors are the Dirac delta functions: ∫δ(x - λ) dx = 1 Thus ∫δ(x - λ)F(x)dx is only defined at x = λ Consider, <λ|ψ(x)> = ∫δ(x-λ)ψ(x) dx = ψ(λ)∫δ(x - λ) dx = ψ(λ) Replace λ with x since ψ only has a value at x <x|ψ> = ψ(x) - component of ψ along x This is the expression of ψ in the x coordinate (Definition 3. from above). Momentum Operator ----------------- Let now look at the differential of ψ w.r.t. x. We write: p = -i∂/∂x|ψ> The factor of -i is required to make the differential hermitian. This can be seen as follows: <ψ|-i∂/∂x|ψ> = ∫ψ*(-i∂/∂x)ψ = -i∫ψ*(∂ψ/∂x) The complex conjugate of this is i∫ψ(∂ψ*/∂x) Apply integation by parts: ∫vdu = uv - ∫udv u = ψ ∴ du = ∂ψ/∂x dv = (∂ψ*/∂x) ∴ v = ψ* [ψψ*] + i∫ψ(∂ψ*/∂x) -∞ -∞ 0 - ∫udv => i∫ψ(∂ψ*/∂x) (Here we are making the assumption that ψ -> 0 at ∞.) This gives: i∫ψ(∂ψ*/∂x) The complex conjugate of this is -i∫ψ*(∂ψ/∂x) which is the integral we started with. Therefore, -i∂/∂x is its own complex conjugate proving that it is hermitian. We can now write the eigenvalue equation as: -i∂/∂x|ψ> = k|ψ> This has a solution of the form ψ = exp(ikx) = coskx + isinkx This gives us the first notion of wave-particle duality as proposed by De Broglie. Particle on a Ring ------------------ The Born Rule ------------- The Born rule in QM postulates that the probability that a measurement on a quantum system will yield a given result is proportional to the square of the magnitude of the particle's wavefunction at that point. Therefore, the probability of finding the particle at x, Pr(x) is given by: Pr(x) = N2ψ*(x)ψ(x) = A2|ψ|2 Where N is a normalization factor. Therefore, the probability of finding the particle somewhere is: ∫dx Pr(x) = N2∫dx ψ*(x)ψ(x) = 1 allspace allspace However, if we do this with the above wavefunction, evaluating the integral becomes an impossible task because we cannot pick a finite value for N. Given that we are dealing with things on an atomic scale it seems reasonable to restrict the analysis to a closed loop of length, L and radius, R. For single valued periodic functions, this imposes the following constraint: ψ(x + L) = ψ(x) Thus, exp(ip(x + L)/h) = exp(ipx/h)exp(ipL/h) => exp(ipL/h) = 1 We can write pL/h = 2πm ∴ p = (2πh/L)m We can also write pL/2π = mh or pR = mh ... the angular momentum Therefore, the momentum, p, is quantized and not continuous. Thus, we can write: So we rewrite our integral as: L L ∫dx P(x) = N2∫dx ψ*(x)ψ(x) = 1 0 0 Now, in order to ensure that the probability = 1 we need to NORMALIZE ψ. Therefore, we must have: L N2∫dx = 1 0 Or, N = 1/√L Therefore, we are now in a position to write ψ as: ψm(x) = (1/√L)exp(2πimx/L) or equivalently ψp(x) = (1/√L)exp(ipx/h) Where the subscripts denote the 'type' of the wavefunction. These are the eigenfunctions of ψp(x) and ψm(x) respectively and αp and αm are the eigenvalues associated with those states. Note that these states are orthogonal to each other. Therefore, ∫dx ψmn = δmn The probabity, Prp(x), that a state is occupied is given by: Prp(x) = |αp|2 Likewise, Prm(x) = |αm|2 Computation of αm ----------------- From before we had: αj = ∫ej(x)*ψ(x)dx = ∫exp(-2πimx/L)ψ(x)dx = ∫exp(-2πimx/L)Σαiexp(2πinx/L)dx = Σαi∫exp(-2πimx/L)exp(2πinx/h)dx = αj when i = j = 0 otherwise For i ≠ j, the real and imaginary parts of the exponential oscillate over |i - j| full cycles and give 0 see proof. Example ------- Consider the following wavefunction. ψ = Ncos(6πx/L) L N2∫dx cos2(6πx/L) = 1 0 => N = √(2/L) L αm = ∫dx (1/√L)exp(-2πimx/L)(√(2/L))cos(6πx/L) 0 We could solve this integral, but there is a quicker way to get at the answer! cos(6πx/L) can be written as: cos(6πx/L) = (exp(6πix/L) + (exp(-6πix/L))/2 We can now just compare this with ψ = (1/√L)Σαmexp(2πimx/L) cos(6πx/L) = (exp(6πix/L) + (exp(-6πix/L))/2 Therefore, ψ = √(2/L)(1/2)(exp(6πix/L) + exp(-6πix/L) = √(1/2L)(exp(6πix/L) + exp(-6πix/L) = (1/√2)exp(6πix/L)/√L + (1/√2)exp(-6πix/L)/√L From this we see that m = 3 and α3 = α-3 = 1/√2 Therefore, ψ = (1/√2)[exp(3πix/L)/√L + exp(-3πix/L)/√L] Therefore, ψ3 = exp(3πix/L)/√L and α3 = 1/√2 ψ-3 = exp(-3πix/L)/√L and α-3 = 1/√2 Note that ψ3 and ψ-3 are orthonormal: L ∫ψ33 dx = (1/L)∫exp(-3πix/L)exp(3πix/L) dx = 1 0 L ∫ψ-33 dx = (1/L)∫exp(3πix/L)exp(3πix/L) dx = 0 0 Therefore we can expand ψ in terms of the basis vectors ψ3 and ψ-3. This is the principle of SUPERPOSITION discussed later. Lets now look at the momentum eigenvalues associated with these eigenvectors. -ih∂ψ3/∂x = (-ih)(1/L)3/2(3πi)exp(3πix/L) = (3πh/L)ψ3 Likewise, -ih∂ψ-3/∂x = (-3πh/L)ψ-3 So the momentum eigenvalues are ±3πh/L We could do exactly the samething to get the energy eigenvalues as follows: ∂2ψ3/∂x2 = (1/√L)(3πi/L)2exp(3πix/L)     = -(9π2/L2)(1/√L)exp(3πix/L)     = -(9π2/L232ψ-3/∂x2 = (1/√L)(3πi/L)2exp(-3πix/L)      = (9π2/L2)(1/√L)exp(3πix/L)      = (9π2/L2-3 If we multiply both sides by -h2/2m we get: E3 = (9π2h2/2mL2) E-3 = -(9π2h2/2mL2) Matrix Element -------------- Recall: Oij = ∫ei*Oijej dx Therefore, p11 = <ψ3|-ih∂/∂x|ψ3> L = (1/L)∫exp(-3πix/L)(3πh/L)exp(3πix/L) dx 0 L = (3πh/L2)∫exp(-3πix/L)exp(3πix/L) dx 0 L = (3πh/L2)∫1 dx 0 = 3πh/L p12 = 0 (orthogonal) p22 = <ψ-3|-(h2/2m)∂2/∂x2-3> L = (1/L)∫exp(3πix/L)(-3πh/L)exp(-3πix/L) dx 0 = -3πh/L p21 = 0 (orthogonal) This is a diagonal matrix consisting of the eigenvalues. - - pij = | 3πh/L 0 |    | 0 -3πh/L | - - We can see this in a different way by using the eigenvalue equation and assuming that the original matrix operator was diagonalizable with the eigenvalues being the diagonal elements. - - -  - - - | p11 0 || ψ3 | = p3| ψ3 | | 0 p22 || 0  |   | 0  | - - -  - - - - - - - - - | p11 0 || 0   | = p-3| 0   | | 0 p22 || ψ-3 |   | ψ-3 | - - - - - - Which gives: - - | p11 - p 0 | = 0 | 0 p22 - p | - - Therefore: (p11 - p)(p22 - p) = 0 p11p22 - p11p - pp22 + p2 = 0 (3πh/L)(-3πh/L) - (3πh/L)p + p(3πh/L) + p2 = 0 Therefore, p2 = 9π2h2/L2 Or, p = ±3πh/L Therefore the momentum eigenvalues, p, are the eigenvalues of the p matrix. Likewise, we could do the same thing for the energy eigenvalues. H11 = <ψ3|-(h2/2m)∂2/∂x23> L = (1/L)∫exp(-3πix/L)(9π2h2/2mL2)exp(3πix/L) dx 0 L = 9π2h2/2mL3∫exp(-3πix/L)exp(3πix/L) dx 0 L = 9π2h2/2mL3∫1 dx 0 = 9π2h2/2mL2 H12 = 0 (orthogonal) H22 = <ψ-3|-(h2/2m)∂2/∂x2-3> L = (1/L)∫exp(3πix/L)(-9π2h2/2mL2)exp(-3πix/L) dx 0 = -9π2h2/2mL2 H21 = 0 (orthogonal) This is a diagonal matrix consisting of the eigenvalues which means that the original Hermitian matrix operator was diagonalizable. - - | 9π2h2/2mL2 0 | | 0 -9π2h2/2mL2 | - - Again we could also use: - - -  - - - | H11 0 || ψ3 | = E3| ψ3 | | 0 H22 || 0  |   | 0  | - - -  - - - - - - - - - | H11 0 || 0   | = E-3| 0   | | 0 H22 || ψ-3 |   | ψ-3 | - - - - - - Which gives: - - | H11 - E 0 | = 0 | 0 H22 - E | - - Therefore: (H11 - E)(H22 - E) = 0 H11H22 - H11E - EH22 + E2 = 0 (9π2h2/2mL2)(-9π2h2/2mL2) - 9π2h2/2mL2E + 9π2h2/2mL2 + E2 = 0 Therefore, E2 = 81π4h4/4m2L4 Or, E = ±9π2h2/2mL2 Therefore the energy eigenvalues, E, are the eigenvalues of the H matrix. Spin Wavefunctions ------------------ Up until now we have been talking about SPATIAL wavefunctions, ψ. There is another type of wavefunction called a SPIN wavefunction, χ. A spatial wave function can be considered as a vector in an infinite dimensional complex vector space. In contrast, a spin wave function is a vector in the spin vector space which is finite dimensional. For spin 1/2 particles, the spin space is 2 dimensional and therefore spanned by 2 base vectors - 'up' and 'down'. The combined wavefunction of the system is the tensor product of the spatial space and the spin space i.e. the full wavefunction has spatial part and spin part. ψTOTAL = χ ⊗ ψ The tensor product factorization is only possible if the Hamiltonian can be split into the sum of spatial and spin terms. In other spin and orbital angular momentum are separable. Also factorization is not possible when the particle interacts with an external field (i.e. magnetic field) or couples to a space dependent quantity (i.e. spin orbit coupling). Spatial wavefunctions are solutions to Schrodinger's Equation. The property of spin comes as a result of relativistic effects and is manifest in the Dirac equation in the form of the DIRAC SPINORS. It corresponds to a new degree of freedom associated with the internal angular momentum state of the particle. Here we treat spin with above assumtions and non relativistically. We can write the spin 1/2 wavefunction in general as: χz = α|+> + β|-> Where, - - - - |+> = | 1 | |-> = | 0 | | 0 | | 1 | - - - - - - - - - - χ = α| 1 | + β| 0 | = | α | | 0 | | 1 | | β | - - - - - - <χ|χ> = 1 - - - - | α*+ β*- || α+ | | β- | - - - - α*α(+.+) + α*β(+.-) + β*α(-.+) + β*β(-.-) = 1 α2 + β2 = 1 α2 = |<e1|χ>|2 β2 = |<e2|χ>|2 Now consider the eigenvalue equation: H|ψ> = E|ψ> - - The Hamiltonian is the Hermitian matrix: | h g | | g h | - - The eigenvalues can be determined from: det(H - EI) = 0 - - det| h - E g | = 0 | g h - E | - - This leads to: E1 = h + g - - v1 = | 1 |   | 1 | - - The normalized vector is: |v1| = √[12 + 12] = √2 Therefore, - - v1 = | 1/√2 |   | 1/√2 | - - E2 = h - g - - v2 = | 1 |   | -1 | - - The normalized vector is: |v2| = √[12 + (-1)2] = √2 Therefore, - - v2 = | 1/√2 |   | -1/√2 | - -      - - - - <e1|e2> = e1.e2 = | 1/√2 1/√2 || 1/√2 | = 0      - - | -1/√2 |      - - So the eigenfunctions are orthogonal We can construct the state |+> as a linear combination of the eigenvectors: - - - - - - α|+> = α| 1 | = α| 1/√2 | + α| 1/√2 | | 0 | | 1/√2 | | -1/√2 | - - - - - - - - - - | 1/√2 1/√2 || 1 | = 1/√2 - - | 0 | - - - - - - | 1/√2 -1/√2 || 1 | = 1/√2 - - | 0 | - - - - - - - - 1/√2|+> = 1/√2| 1 | = 1/√2| 1/√2 | + 1/√2| 1/√2 | | 0 | | 1/√2 | | -1/√2 | - - - - - - - - - - - - β|-> = β| 0 | = β| 1/√2 | - β| 1/√2 | | 1 | | 1/√2 | | -1/√2 | - - - - - - - - - - | 1/√2 1/√2 || 0 | = 1/√2 - - | 1 | - - - - - - | 1/√2 -1/√2 || 0 | = -1/√2 - - | 1 | - - - - - - - - 1/√2|-> = 1/√2| 1 | = 1/√2| 1/√2 | + 1/√2| 1/√2 | | 0 | | 1/√2 | | -1/√2 | - - - - - - α2 + β2 = 1 - - - - - - H11 = | 1/√2 1/√2 || h g || 1/√2 | = h + g    - - | g h || 1/√2 | - - - - H12 = 0 H21 = 0 - - - - - - H22 = | 1/√2 -1/√2 || h g || 1/√2 | = h - g    - - | g h || -1/√2 | - - - - Thus, we get: - - | h + g 0 | | 0 h - g | - - This is a real symmetric matrix with diagonal equal to the eigenvalues. Eigenspinors ------------ In the case of a single spin 1/2 particle like the elecron, eigenspinors are eigenvectors of the Pauli matrices. Each set of eigenspinors forms a complete, orthonormal basis. This means that any state can be written as a linear combination of the basis spinors. Each of the (Hermitian) Pauli matrices has 2 eigenvalues, +1 and -1. The corresponding normalized eigenspinors are: - - - - - - σx = | 0 1 | with |+> = 1/√2| 1 | and |-> = 1/√2| -1 |   | 1 0 | | 1 | | 1 | - - - - - - - - - - - - σy = | 0 -i | with |+> = 1/√2| -i | and |-> = 1/√2| i |   | i 0 | | 0 | | 1 | - - - - - - - - - - - - σz = | 1 0 | with |+> = | 1 | and |-> = | 0 |   | 0 -1 | | 0 | | 1 | - - - - - - Therefore, when we wrote: χz = α|+> + β|-> Where, - - - - |+> = | 1 | |-> = | 0 | | 0 | | 1 | - - - - We were writing that our general state is a linear combination of the normalized eigenspinors of the σz matrix. We could have picked the eigenspinors for the x or y directions as a basis, but the z direction is chosen by convention. It is worth noting that if we had chosen h = 0 and g = 1 in the above analysis, it would be equivalent to using the σx matrix as the operator. Matrix Element: Consider the 2 states: |1> = √(1/2)(|-> - i|+>) with α2 + β2 = 1 |2> = √(1/2)(|-> + i|+>) with α2 + β2 = 1 - - - - |j> = |+> = | 1 | and |-> = | 0 | | 0 | | 1 | - - - - - - - - |1> = √(1/2)| 0 | - 1√(1/2)| 1 | | 1 | | 0 | - - - - = √(1/2)| -i | | 1 | - - Similarly, - - - - |2> = √(1/2)| 0 | + 1√(1/2)| 1 | | 1 | | 0 | - - - - = √(1/2)| i | | 1 | - - - - - - <i| = <+| = | 1 0 | and <-| = | 0 1 | - - - - - - <1| = √(1/2)| i 1 | - - - - <2| = √(1/2)| -i 1 | - - Nww consider an operator, S, that does the following: S|+> = (+ih/2)|-> and S|-> = (-ih/2)|+> The matrix elements are found from: - - - - | <+|S|+> <+|S|-> | -> | (ih/2)<+|-> (-ih/2)<+|+> | | <-|S|+> <-|S|-> | | (ih/2)<-|-> (ih/2)<-|+> | - - - - - - = | 0 -ih/2 | | ih/2 0 | - - Therefore, S = S2 = (h/2)σ2 The eigeenvalues of S are given by: S|1> = λ|1> - - - - = | 0 -ih/2 || -i√(1/2) | | ih/2 0 || √(1/2) | - - - - - - = (-ih/2)| √(1/2) | | i√(1/2) | - - - - = (h/2)| -i√(1/2) | | √(1/2) | - - Quantum Superposition --------------------- Quantum superposition is a fundamental principle of QM that says we can expand any wavefunction in terms of a complete set of basis vectors that correspond to the eigenstates of the system. The physical system exists partly in all of these possible eigenstates simultaneously but, when observed, it gives a result corresponding to only one of the possible configurations. This is referred to as the 'collapse' of the wave function. Summarizing: - Initially, the wave function is a superposition of the possible eigenstates (eigenvectors). - After measurement (interaction with an observer) the wave function collapes into a single eigenstate. Therefore, collapse Superposition of eigentates => single eigenstate For the spatial wavefunction: ψ(x) = ΣAmψ(x) => Amexp(ipmx/h) For the spin 1/2 system the wavefunction: χz = α|+> + β|-> => |+> or |-> Where α2 + β2 = 1 The notion of a wavefunction collapse can be seen in the famous Schrodinger's cat paradox. A cat is placed in an opaque chamber with a device containing cyanide that can be released by a mechanism triggered by the radioactive decay of a minute amount of Uranium (also placed inside the chamber). Since the decay process is random, an observer does not know if the cat is dead or alive at any given moment. In QM this situation represents the superposition of the 2 possible states. When the chamber is opened, the superposition is lost, and the cat is seen to be either dead or alive. Thus, the observation or measurement itself affects an outcome, so that the outcome as such does not exist unless the measurement is made. (That is, there is no single outcome unless it is observed.) Summary of QM Postulates ------------------------ 1. States of a system are represented by vectors (functions) in a complex vector space (HILBERT SPACE). 2. Observables are represented by Hermitian operators. 3. The values that the observable can have are eigenvalues of the Hermitian operator. 4. States that have definite (certain) values are the eigenvectors.. 5. If system prepared in state |ψ(x)> (not an eigenvector) then the probability of finding eigenvalues λ with corresponding eigenvectors |λ> is: P(λ) = |Aλ|2 Where Aλ = ∫ψm(x)*ψ(x)dx and ψm(x) = exp(2πimx/L) Appendix -------- Fourier Series Using Complex Exponentials ----------------------------------------- The Fourier series is used to represent a periodic function, f(x) by: f(x) = a0 + Σancosωnx + bnsinωnx n=1 Where, T a0 = (1/T)∫f(t)dt 0 T an = (2/T)∫dt f(t)cosωmt 0 T bn = (2/T)∫dt f(t)sinωmt 0 We can introduce complex exponentials as follows: cosωnx = [exp(iωnx) + exp(-iωnx)]/2 and sinωnx = [exp(iωnx) - exp(-iωnx)]/2i Therefore, f(x) = a0 + Σan[exp(iωnx) + exp(-iωnx)]/2 - ibn[exp(iωnx) - exp(-iωnx)]/2 n=1 = a0 + Σ[(an - ibn)/2]exp(iωnx) + [an + ibn)/2]exp(-iωnx)] n=1 = a0 + Σ[cnexp(iωnx) + c-nexp(-iωnx)] n=1 Where c0 = a0 cn = (an - ibn)/2 c-n = (an + ibn)/2 We can write the above as: +∞ f(x) = Σcnexp(iωnx) -∞ T/2 Where cn = (1/T)∫dx f(x)exp(-iωnx) -T/2 Proof ("Fourier's trick"): T/2 +∞ T/2 ∫dx f(x)exp(-iωmx) = Σ ∫dx cnexp(iωnx)exp(-iωmx) -T/2 -∞ -T/2 = cnT Since, T/2 ∫dx exp(iωnx)exp(-iωmx) = δmn (orthogonality theorem) -T/2 Therefore, T/2 cn = (1/T)∫dx f(x)exp(-iωnx) -T/2 Fourier Series to Fourier Transform ----------------------------------- The Fourier transform is an extension of the Fourier series that results when the function, f(x), vanishes outside of the interval [-T/2,T/2]. In other words, when the period of the represented f(x) is lengthened and allowed to approach infinity. The Fourier transform is then defined by: T/2 f(ω) = ∫dx f(x)exp(-iωx) = lim ∫dx f(x)exp(-iωx) -∞ T->∞ -T/2 Comparing this to the definition of the Fourier transform, it follows that: cn = (1/T)f(ωn) = (1/T)f(n/T) Where nω = ωn = n/T <- Δω -> -------+--------+--------> T n/T (n + 1)/T So, Δω = (n + 1)/T - n/T = 1/T f(x) = lim Σcnexp(i(n/T)x) T->∞ -∞ = lim Σ(1/T)f(n/T)exp(i(n/T)x) T->∞ -∞ = lim ΣΔω f(ωn)exp(iωnx) Δω->0 -∞ This is the Riemann sum that approximates an integral by a finite sum. Therefore, we can write: f(x) = ∫dω f(ω)exp(iωx) -∞ For a Fourier series, cn can be thought of as the amount of the wave present in the Fourier series of f(x). Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of each individual frequency is present in f(x), and these waves can be recombined by using an integral (or 'continuous sum') to reproduce the original function. In summary, +∞ Therefore, f(x) = ∫dω f(ω)exp(iωx) is just the -∞ continuous analog of f(x) = Σcnexp(iωnx) -∞ Normalization ------------- The normalization factor, N, has to satisfy: N2∫dx |f(x)|2 = 1 allspace Fourier Series/Transform in QM/QFT ---------------------------------- The normalization factor, N, has to satisfy: N2∫dx ψ*(x)ψ(x) = 1 allspace For eigenfunctions this becomes: L N2∫dx exp(-ikx)exp(ikx) = 1 0 L N2∫dx = 1 0 L N2|x| = N2L = 1 ∴ N = 1/√L 0 For a particle on a ring, L = 2π. Therefore: N = 1/√(2π) Now we know that exp(ikx) and exp(-ikx) have normalization factors of 1/√(2π) we can therefore write the Fourier transforms of the position and momentum states as: ψ(x) = (1/√(2π))∫dk φ(k)exp(ikx) -∞ φ(k) = (1/√(2π))∫dx ψ(x)exp(-ikx) -∞ Note: In calculations φ(x) and ψ(x) also need to be normalized to get the correct result. We can write: ψ(x) = (1/√(2π))∫dk φ(k)exp(ikx) -∞ = (1/√(2π))∫dk exp(ikx) (1/√(2π))∫dx' ψ(x')exp(-ikx') -∞ -∞ = ∫dx' ψ(x') (1/2π)∫dk exp(ik(x - x')) -∞ -∞ = ∫dx' ψ(x')δ(x - x') -∞ = ψ(x) Where δ(x - x') is the Dirac Delta function. Examples -------- Ex 1. Consider ψ(x) = cos(6πx/L) N2∫dx ψ*(x)ψ(x) = 1 allspace L N2∫dx (cos(6πx/L))*cos(6πx/L) = 1 0 L N2∫dx cos2(6πx/L) = 1 0 N2L/2 = 1 ∴ N = √(2/L) = √(1/π) Now k = 2π/λ and L = nλ so k = 2πn/L and 6πx/L ≡ 3k. Therefore, (1/√(2π))∫dx √(1/π)cos(3kx)exp(-iknx) -∞ = (1/√(2π))∫dx (1/2){√(1/π)exp(i3k) + √(1/π)exp(-i3k)}exp(-iknx) = (1/√(2π))√(1/π)∫dx (1/2){exp(i(3 - kn)x + exp(-i(3 + kn)x} = (1/√(2π))√(1/π)∫dx (1/2){exp(-i(kn - 3)x + exp(-i(3 + kn)x} = (1/√(2π))√(1/π)(1/2)2π{δ(kn - 3) + δ(kn + 3)} = (1/√2){δ(kn - 3) + δ(kn + 3)} Therefore, ψ(x) can be written as: ψ(x) = Σcnexp(iknx) = (1/√2)exp(ik3x) + (1/√2)exp(-ik3x) -∞ = (1/√2)|exp(ik3x)> + (1/√2)|exp(ik-3x)> When a measurement is made, ψ(x) will collapse to the eigenstate with either kn = +3 or kn = -3 with equal probability (= (1/√2)2 = 1/2). No other possibilites are allowed. Ex 2. Consider ψ(x) = exp(-α|x|). This looks like: N2∫dx exp(-α|x|)*exp(-α|x|) = 1 N2∫dx exp(-2α|x|) = 1 0 ∫dx exp(-2a(-x)) + ∫dx exp(-2a(+x)) -∞ 0 0 = |exp(-2a(-x))/2α| + |exp(-2a(+x))/-2α| -∞ 0 = [1/2α - 0] + [0 + 1/2α] = N2/α = 1 ∴ N = √α ∫dx (√α)exp(-α|x|)(1/(√2π)))exp(-ikx) -∞ 0 = √(α/2π)[∫dx exp(αx)exp(-ikx) + ∫dt exp(-αx)exp(-ikx)] -∞ 0 0 = √(α/2π)|exp(x(α - ik)/(α - ik) | + |exp(x(-α - ik)/(-α - ik) | -∞ 0 = √(α/2π)[1/(α - ik) - 0 + 0 - 1/(-α - ik)] = √(α/2π)[(α + ik) + (α - ik)]/(α + ik)(α - ik) = √(α/2π)2α/(α2 + k2) ψ(x) = Σcnexp(ikx) = Σ√(α/2π)(2α/(α2 + k2))|exp(iknx)> -∞ -∞ This means every k a possible result in a momentum measurement, since |cn|2 does not vanish for any (finite) k. However, the maximum probability corresponds to k = 0. Potential Well -------------- ------------ --------- V(x) | | | | | | --------- 0 L Inside the well, V(x) = 0 The time independent Schrodinger equation is: ∂2ψ/∂x2 + 2mEψ/h2 = 0 or, ∂2ψ/∂x2 = -2mEψ/h2 = -k2ψ Where k = √(2mE)/h This has the solution: ψ(x) = Asinkx + Bcoskx ≡ Cexp(ikx) + Dexp(-ikx) The boundary conditions are: ψ(0) = ψ(L) = 0 ψ(0) = Asink0 + Bcosk0 = 0 => B = 0 and, ψ(L) = AsinkL = 0 => A = 0 or sinkL = 0 In the latter case, this implies that k = nπ/L Now, E = p2/2m = h2k2/2m = n2π2h2/2mL2 We now find A by normalizing ψ(x): L A2∫dx sin2(nπx/L) = 1 0 A2L/2 = 1 ∴ A = √(2/L) ψ(x) = √(2/L)sin(nπx/L) L cn = (2/L)∫dx f(x)sin(mπx/L) 0 L = √(2/L)√(2/L)∫dx sin(nπx/L)sin(mπx/L) 0 sinxsiny = (1/2){cos(x - y) - cos(x + y)} L = (2/L)(1/2)∫dx {cos(πx(n - m)/L) - cos(πx(n + m)/L)} 0 L = (2/L)(L/2)|sin(πx(n - m)/L)/(n - m)π - sin(πx(n + m)/L)/(n + m)π| L 0 = |sin(πx(n - m)/L)/(n - m)π - sin(πx(n + m)/L)/(n + m)π| 0 = {sin(π(n - m))/(n - m)π - sin(π(n + m))/(n + m)π} = 0 for n ≠ m L (2/L)∫dx sin2(nπx/L) = 1 0 ψ(x) = Σcn√(2/L)sin(nπx/L) -∞ The negative solutions give nothing new since sin(-nπx/L) = -sin(nπx/L) and the sign can be absorbed in the constant. Therefore, ψ(x) = Σ√(2/L)sin(nπx/L) n=1 Now, √(2/L)sin(nπx/L) = √(2/L)(1/2i)(exp(inπx/L) + (exp(-inπx/L) = (-i/√2)(exp(inπx/L)/√L + (exp(-inπx/L)/√L ih∂/∂x|ψ(x)> = p|ψ(x)> gives, ih∂/∂x|(-i/√2)(exp(inπx/L)/√L> = (-i/√2)inπ/√L|(exp(inπx/L)/√L> = (1/√2)nπ/L Similarly, ih∂/∂x|(-i/√2)(exp(-inπx/L)/√L> = (-i/√2)-inπ/√L|(exp(-inπx/L)/√L> = -(1/√2)nπ/L This says that the probability of the particle having a momentum of nπ/L is the same for both the +x and -x direction. We can also compute the expectation value of p as: L <p> = (2/L)∫dx <ψ|-ih∂/∂x|ψ> 0 L = (2/L)∫dx <sin(nπx/L)|ih∂/∂x|sin(nπx/L)> 0 L = (2/L)∫dx sin(nπx/L)(nπx/L)cos(nπx/L) 0 = 0 Note: This does not mean that the kinetic energy is 0!