Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

-
Celestial Coordinates
-
Celestial Navigation
-
Distance Units
-
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

-
Penrose Diagrams
-
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
-
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

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 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 Qubit
-
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

test

-
test

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

Pauli Spin Matrices ------------------- The Pauli matrices are related to the angular momentum operator that corresponds to an observable describing the spin of a spin 1/2 particle. They are Hermitian and unitary. The Pauli matrices (after multiplication by i to make them anti-Hermitian), also generate transformations in the sense of Lie algebras The Pauli spin matrices have eigenvalues of +1 and -1. Thus, sx,y,z|χ> = ±|χ> where χ is the spin wavefunction. (h/2)σx,y,z|χ> = ±|χ> We can represent χ as a linear combination of 'up' and 'down' states as follows: χ = α|up> + β|down> z-axis component: - - - - - - |1 0|| α | = ±| α | |0 -1|| β | | β | - - - - - - Positive: - - - - | α | = +| α | | -β | | β | - - - - ∴ β = -β so β must equal 0 and - - - - | α | = +| α | | -β | | 0 | - - - - Negative: - - - - | α | = -| α | | -β | | β | - - - - ∴ α = -α so α must equal 0 and - - - - | α | = -| 0 | | -β | | β | - - - - Therefore, the eigenvectors are, - - - - | 1 | and | 0 | | 0 | | 1 | - - - - x-axis component: - - - - - - |0 1|| α | = ±| α | |1 0|| β | | β | - - - - - - Positive: - - - - | β | = +| α | | α | | β | - - - - ∴ β = α so - - - - | β | = +| α | | α | | α | - - - - Negative: - - - - | β | = -| β | | α | | α | - - - - ∴ β = -α so - - - - | β | = -| α | | α | | -α | - - - - Now, |α|2 + |β|2 = 1 ∴ 2α2 = 1 so α = β = 1/√2 Therefore, the eigenvectors are, - - - - |1/√2| and | 1/√2| |1/√2| |-1/√2| - - - - y-axis component: - - - - - - |0 -i|| α | = ±| α | |i 0|| β | | β | - - - - - - Positive: - - - - | -iβ | = +| α | | iα | | β | - - - - ∴ -iβ = α so β = iα and - - - - | -iβ | = +| α | | iα | | iα | - - - - Negative: - - - - | -iβ | = -| α | | iα | | β | - - - - ∴ -iβ = -α so β = -iα and - - - - | -iβ | = -| α | | iα | | -iα | - - - - Therefore, the eigenvectors are, - - - - |1/√2| and | 1/√2| |i/√2| |-i/√2| - - - - Since Sx, Sy, Sz must have eigenvalues of +/-h/2, the σ matrices must have eigenvalues of +/-1. S2 = Sx2 + Sy2 + Sz2 Sx = (h/2)σx Sy = (h/2)σy Sz = (h/2)σz S2 = Sx2 + Sy2 + Sz2 |S|2 = (h2/4){σx2 + σy2 + σz2} - - - - - - = (h2/4){| 1 0 | + | 1 0 | + | 1 0 |}   | 0 1 | | 0 1 | | 0 1 | - - - - - - - - = (3h2/4)| I | - - - - = s(s + 1)h2| I | where s = 1/2 - - This should be compared with the equivalent relation for the orbital angular momentum operator that is obtained when solving the Schrodinger equation for the hydrogen atom. |L|2 = l(l + 1)h2