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Astronomy

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Celestial Coordinates
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Celestial Navigation
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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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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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Gravity - Force and Acceleration
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Kinematics
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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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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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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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Planck Units
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Finance and Accounting

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General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Einstein's Field Equations
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Tensors
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Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Mathematics

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Complex Numbers
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Integration By Parts
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Introduction to Conformal Field Theory
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Vector Calculus
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Volume Integrals

Microeconomics

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Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Spin 1 Eigenvectors
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Probability and Statistics

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Programming and Computer Science

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Hashing
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How this site works ...
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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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Commutators
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Dyson Series
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Photoelectric Effect
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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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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: January 26, 2018

Inflation Theory --------------- Inflation theory proposes a period of extremely rapid (exponential) expansion of the universe between 10-36 and 10-32 seconds after the Big Bang followed by a more gradual expansion. During this time the universe expanded ~ e60 times. For inflation to have occurred it is postulated that some scalar field exists with a particular potential function such that when its vacuum type energy density is very large that density decreases very slowly. When its energy density decreases past a certain point, it stops behaving in this way and starts decreasing at the same rate as ordinary matter. Such a field is referred to as the 'INFLATON' field, φ. φ can be represented by: Potential Energy V(φ) slight negative slope | | | V | A------------B | \ | \ <-- Big Bang (starting heat energy (aka reheating) | \ | \ | -- <- Λ (Dark energy) ------------------------------ φ We can imagine the universe as a ball slowly rolling down the slope and then falling over a 'cliff' where the ball literally represents the value of the field at any particular point. The energy of the field per unit volume is given by: . E = φ2/2 + (∇φ)2 + V(φ) For a homogeneous and isotropic field ∇φ = 0. We can write: . Lagrangian: L = [φ2/2 - V(φ)]a3 . Euler-Lagrange: d(a3φ/2)/dt = -a3∂V/∂φ .. . . => φ + 3(a/a)φ = -∂V/∂φ . The 3(a/a) term is analagous the 'friction' term in the viscosity equation above. We can write: .. . φ + 3Hφ = -∂V/∂φ This is the HUBBLE FRICTION equation. Returning to the FRW equation: H2 = 8πGρ/3 .  = 8πG(φ2/2 + V(φ))/3 Assume that φ is moving very slowly so we can write: H2 = 8πGV(φ)/3 . (a/a) = √(8πGV(φ)/3) This has a solution: a = c'exp{√(8πGV(φ)/3)t} ~ e60! So during the inflationary period the universe expanded enormously fast. Any 'wrinkles' got flattened out and space becomes very homogeneous and isotropic. Consider a radiation dominated universe with k = +1: H2 = (8/3)πMG/a4 - k/a2. During inflation the energy density is enormously large and decreases very slowly. If the scale factor increases extremely rapidly, the (8/3)πMG/a4 will dominate the k/a2 term despite the fact that there is a factor of 100 difference in their denominators. Essentially, the k/a2 term becomes insignificant which is consistant with what we observe today. Another way to look at the flatness issue is to consider a small ballon. An ant on the surface would see a certain amount of curvature. If that balloon is expanded to the size of the Earth, the ant would see the surface as being flat in all directions. Likewise any 'lumpiness' on the surface of the balloon would get smoothed out in the process. The heights of the lumps won't change with respect to the surface (technically not true but good enough to illustrate the idea) but their widths will. As a result, their inclines decrease giving the appearance of a flatter surface. Inflation also successfully explains the absence of relic particles. Inflation allows for magnetic monopoles to exist as long as they were produced prior to the period of inflation or during its early stages when temperatures were hot enough. During inflation, the density of monopoles drops exponentially, so their abundance has dropped to undetectable levels. It is worth noting that since inflation reduces the particle density to virtually zero, we know the particles that exist in the universe today must all have been produced after the inflation epoch. Finally, since inflation supposes a burst of exponential expansion in the early universe, it follows that distant regions were actually much closer together prior to inflation than they would have been with only a standard Big Bang expansion. Thus, such regions could have been in contact prior to inflation and could have attained a uniform temperature. Thus, inflation also solves the 'horizon problem'. Damped Oscillator ------------------- The damped oscillator is described by the equation: .. . mx + ηx + ω2x = 0 Where ω is the restoring force. This has solutions of the form x = Aexp(αt). Substituting this with m = 1 gives: (α2 + ηα + ω2) = 0 (the exp(αt) term is ignored) Therefore, α = -η/2 ± √(η2 - 4ω2)/2 There are 3 cases: Underdamped: η2 < 4ω2 With solution: x = Aexp{-ηt/2)exp[±i√(4ω2 - η2)/2]t} Critically Damped: η2 = 4ω2 or η = ±2ω Overdamped: η2 > 4ω2 It is the underdamped and critically damped solutions that are of the most interest. Recall, the wave equation for a scalar field in flat spacetime is (c = 1): ∂2φ/∂t2 - ∂2φ/∂x2 = -∂V/∂φ For the inflaton field we write: ∂2φ/∂t2 - (1/a2)∂2φ/∂x2 = 0 Note the 1/a2 term is needed because of the stretching of space. We can rewrite the Hubble Friction equation as: ∂2φ/∂t2 + 3H∂φ/∂t - (1/a2)∂2φ/∂x2 = 0 With solution φ = φ(t)exp(ik'x) where k' = 2πa/λ Thus, we get: .. . φ + 3Hφ + (k'/a)2φ = 0 (the exp(ik'x) term is ignored) This has exactly the same form as the damped oscillator equation where k'/a has taken the place of ω and η = 3H So what this means is that waves that start out with different values of k' get stretched as the universe expands. They go from underdamped to overdamped and then 'freeze' when the restoring force, k'/a, becomes 0. The point at which the transition between underdamped and overdamped (critical damping) occurs when η = 2ω or η = 2k'/a. Thus, 3H = 2k'/a or a = 4πa/3Hλ after substituting for k' = 2πa/λ For this equation to be true λ must equal: λ = 4π/3H The distance at which they get frozen corresponds to 4π/3H - the HUBBLE HORIZON. Thus, the shorter λ is the longer it takes for the wave to stretch to the current horizon and freeze and vice versa. The point is that, regardless of its λ, a wave can only stretch as far as the current horizon before it gets frozen. Once critical damping takes place, the restoring force starts to disappear. The equation for the damped oscillator from before becomes: (α2 + ηα) = 0 Or, α(α + η) = 0 Which has the solution: A0 + A1exp(-ηt) Thus, the wave exponentially decays and freezes at the value A0. So what is the source of these microscopic waves or oscillations in the inflaton field (inflaton waves) that eventually get frozen at the horizon? The answer comes from quantum mechanics. As the universe effectively 'rolls' down the slope of the field, there are quantum fluctuations in the vacuum that continously generate microscopic waves of different wavelengths. These waves superpose on top of each other with each one eventually undergoing the process described above. Schematically, we can look at this in the following manner. Consider an artificial 'overhead' view. B = top of the cliff x | | v | | + | \X + X' | / + | Y / + Y' | \ + | | + -----------------------> φ -----------------------> t, a If there were no fluctuations in the vacuum, the 'zig-zag' line would just be straight. X and Y would fall over the cliff at the same time and their subsequent densities would decrease at the same rate. If, however, we now add the effect of quantum fluctuations we get a different result. Now the line is blurred ('zig-zag'). In this case, X goes over the cliff first and slides down the hill where reheating occurs. X becomes X' and moves off with decreasing density as the universe continues to expand. Y follows X to form Y' but Y' is delayed in time so his density will be different to X'. This process created inhomogenities in the universe that eventually led to the higher density regions condensing into the stars, galaxies, and clusters of galaxies that we see today.