Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

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

Chemistry

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

Classical Physics

-
Archimedes Principle
-
Bernoulli Principle
-
Blackbody (Cavity) Radiation and Planck's Hypothesis
-
Center of Mass Frame
-
Comparison Between Gravitation and Electrostatics
-
Compton Effect .
-
Coriolis Effect
-
Cyclotron Resonance
-
Dispersion
-
Doppler Effect
-
Double Slit Experiment
-
Elastic and Inelastic Collisions .
-
Electric Fields
-
Error Analysis
-
Fick's Law
-
Fluid Pressure
-
Gauss's Law of Universal Gravity .
-
Gravity - Force and Acceleration
-
Hooke's law
-
Ideal and Non-Ideal Gas Laws (van der Waal)
-
Impulse Force
-
Inclined Plane
-
Inertia
-
Kepler's Laws
-
Kinematics
-
Kinetic Theory of Gases .
-
Kirchoff's Laws
-
Laplace's and Poisson's Equations
-
Lorentz Force Law
-
Maxwell's Equations
-
Moments and Torque
-
Nuclear Spin
-
One Dimensional Wave Equation .
-
Pascal's Principle
-
Phase and Group Velocity
-
Planck Radiation Law .
-
Poiseuille's Law
-
Radioactive Decay
-
Refractive Index
-
Rotational Dynamics
-
Simple Harmonic Motion
-
Specific Heat, Latent Heat and Calorimetry
-
Stefan-Boltzmann Law
-
The Gas Laws
-
The Laws of Thermodynamics
-
The Zeeman Effect .
-
Wien's Displacement Law
-
Young's Modulus

Climate Change

-
Keeling Curve .

Cosmology

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

Finance and Accounting

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

Game Theory

-
The Truel .

General Relativity

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

Lagrangian and Hamiltonian Mechanics

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

Macroeconomics

-
Lecture Notes on International Economics
-
Lecture Notes on Macroeconomics
-
Macroeconomic Policy

Mathematics

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

Microeconomics

-
Marginal Revenue and Cost

Particle Physics

-
Feynman Diagrams and Loops
-
Field Dimensions
-
Helicity and Chirality
-
Klein-Gordon and Dirac Equations
-
Regularization and Renormalization
-
Scattering - Mandelstam Variables
-
Spin 1 Eigenvectors .
-
The Vacuum Catastrophe

Probability and Statistics

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

Programming and Computer Science

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

Quantum Computing

-
The Qubit .

Quantum Field Theory

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

Quantum Mechanics

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

Semiconductor Reliability

-
The Weibull Distribution

Solid State Electronics

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

Special Relativity

-
4-vectors .
-
Electromagnetic 4 - Potential
-
Energy and Momentum, E = mc2
-
Lorentz Invariance
-
Lorentz Transform
-
Lorentz Transformation of the EM Field
-
Newton versus Einstein
-
Spinors - Part 1 .
-
Spinors - Part 2 .
-
The Lorentz Group
-
Velocity Addition

Statistical Mechanics

-
Black Body Radiation
-
Entropy and the Partition Function
-
The Harmonic Oscillator
-
The Ideal Gas

String Theory

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

Superconductivity

-
BCS Theory
-
Introduction to Superconductors
-
Superconductivity (Lectures 1 - 10)
-
Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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

The Standard Model

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

Topology

-

Units, Constants and Useful Formulas

-
Constants
-
Formulas
Last modified: January 26, 2018

Isospin ------- Isospin was originally introduced to explain symmetries of the neutron. Protons and neutrons have roughly the same mass. If we neglect their charge (a resonable assumption given that the charge is small in comparison to other forces inside the nucleus), we can consider them to be symmetric although the symmetry is not exact. Similar to a spin 1/2 particle, which have 2 states, protons and neutrons were said to be of isospin, 1/2. The proton and neutron were then associated with different isospin projections I3 = +1/2 and -1/2 respectively where I3 is te component of isospin along the z-axis. With the discovery of quarks, isospin was extended to include the up and down quark since both are very similar in mass, and have the same strong interactions. Particles made of the same numbers of up and down quarks have similar masses and are grouped together. After the quark model was elaborated, it was found that the isospin projection was related to the up and down quark content of particles. The relation is: I3 = (1/2){(nu - nu) - (nd - nd)} Where the underscore represents the antiparticle. Thus, I3 of u is +1/2 and the I3 of d is -1/2. I3 is a quantum number related to the strong interaction. It is NOT spin in the sense of spin angular momentum but does follow the same mathematics i.e. SU(2). For example, a proton-neutron pair can be coupled in a state of total isospin 1 or 0. Thus, we can use the same diagrams as spin angular momentum when we talk about isospin. | | | + +3/2 | | + +1 | | | | | | + +1/2 | + +1/2 | | | | + 0 | + 0 | | | | | | + -1/2 | + -1/2 | | | | | | | | | | + -1 + -3/2 p and n Pions Deltas The number of spin/isospin states is given by: nstates = (2l + 1) where l is the spin/isospin Extention to SU(3) - Flavor Symmetry ------------------------------------ With the discovery of new particles there seemed to be an enlarged symmetry that contained isospin as a subgroup. This larger symmetry was named the Eightfold Way whose origins were explained by up, down and strange quarks which would belong to the fundamental representation of the SU(3) flavor symmetry. The SU(2) symmetry is slightly broken because the u and d quark masses are slightly different. However, the SU(3) symmetry is badly broken due to the much higher mass of the strange quark. Gell-Mann–Nishijima Formula --------------------------- The GMN formula relates all flavour quantum numbers (isospin up and down, strangeness, charm, bottomness, and topness) with the baryon number and the electric charge. Q = I3 + (1/2)(B + S) where: B = (1/3)(nq - nq) = (# of quarks - # of antiquarks) B is the baryon number and S is the strangeness quantum number. Thus, For an up quark we get: Q = 1/2 + (1/2)(1/3) = 2/3 Hypercharge ----------- Hypercharge, Y, is a quantum number relating the charge, Q, and the isospin, I3. Mathematically, hypercharge is: Y = S + C + B' + T + B Where, S = Strangeness C = Charm B' = Bottomness T = Topness B = Baryon number Isospin, electric charge and hypercharge are related by. Q = I3 + Y/2 I3 Q Baryon S Y Mass (MeV) --- --- ------ --- --- --------- u 1/2 2/3 1/3 0 1/3 1.7 - 3.3 d -1/2 -1/3 1/3 0 1/3 4.1 - 5.8 s 0 -1/3 1/3 -1 -2/3 101 Weak Isospin ------------ Consider the decay of a neutron to a proton inside an atomic nucleus that increases the atomic number by 1, while emitting an electron and an anti-neutrino. _ udd -> uud + e + νe u e \ / \ / \/\/\/\/\/\/\/ / W- \ / \ _ / \ νe d If we were to look at this in terms of isospin we would not be able to balance the equation because the isospin only connects quarks and not leptons. In order to fix this we need to introduce the concept of weak isospin, I3, that applies to all particles. The Weak Isospin serves as a quantum number and governs how that particle behaves in the weak interaction. Like isospin, Weak Isospin is described by the mathematics of SU(2). In any given weak interaction, weak isospin is conserved meaning that the sum of the weak isospin numbers of the particles entering the interaction equals the sum of the weak isospin numbers of the particles exiting the interaction. Unlike regular isospin symmetry, which is only approximate, weak isospin symmetry is exact. Weak Hypercharge ---------------- The weak hypercharge, YW, is a quantum number relating the electric charge, Q, and the weak isospin, I3. The weak hypercharge unifies weak interactions with electromagnetic interactions. The weak hypercharge is associated with with the U(1)Y component of SU(2)L ⊗ U(1)Y. It is different but related to the electric charge associated with the U(1)EM of QED. It satisfies the equality: Q = I3 + YW/2 ∴ YW = 2(Q - I3) where Q is in units of e. Q, I3 and YW for left-handed fermions is: Q I3 YW --- --- ---- νe 0 1/2 -1 e -1 -1/2 -1 u 2/3 1/2 1/3 d -1/3 -1/2 1/3 s -1/3 -1/2 1/3 Q, I3 and YW for right-handed fermions is: Q I3 YW --- --- ---- νe does not exist e -1 0 -2 u 2/3 0 4/3 d -1/3 0 -2/3 s -1/3 0 -2/3 Q, I3 and YW for the gauge bosons is: Gauge bosons: Q I3 YW --- --- ---- Z0 0 0 0 W+ 1 1 0 W- -1 -1 0 Particles with weak hypercharge interact by exchange of a boson, called the B boson. The B boson is very similar to more familiar U(1) gauge boson ... the photon. Thus the weak hypercharge force is a lot like electromagnetism but its strength is proportional to the weak hypercharge rather than the charge. Experimentally, it is observed that only left handed (aka left-chiral) fermions and right-handed anti-fermions (aka right-chiral) participate in the weak interaction. Consider: _ uLdLdL -> uLuLdL + eL + νR The weak isospins, I3, are: (1/2 - 1/2 - 1/2) = (1/2 + 1/2 - 1/2) - (1/2) - (1/2)* ∴ (-1/2) = (1/2) - (1/2) - (1/2) ∴ (-1/2) = (-1/2) and the weak isospin is conserved. _ * νR ≡ -νL (note that νR's have yet to be observed directly in nature) The weak hypercharges, YW, are: (1/3 + 1/3 + 1/3) = (1/3 + 1/3 + 1/3) - 1 + 1 ∴ 1 = 1 and the weak hypercharge is conserved. The charges, Q, are: (2/3 - 1/3 - 1/3) = (2/3 + 2/3 - 1/3) - 1 + 0 ∴ 0 = 0 and the charge is conserved. Note: I3 and YW are not conserved when the symmetry is spntaneously broken. This is discussed in detail in the sections on electroweak unification and the Higgs mechanism.