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 Mechanics

-
Blackbody Radiation .

Classical Physics

-
Archimedes Principle
-
Bernoulli Principle
-
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
-
Maxwell's Equations .
-
Moments and Torque
-
Nuclear Spin
-
One Dimensional Wave Equation .
-
Pascal's Principle
-
Phase and Group Velocity
-
Poiseuille's Law
-
Refractive Index
-
Rotational Dynamics
-
Simple Harmonic Motion
-
Specific Heat, Latent Heat and Calorimetry
-
The Gas Laws
-
The Laws of Thermodynamics
-
The Zeeman Effect .
-
Young's Modulus

Climate Change

-
Keeling Curve .

Cosmology

-
Baryogenesis
-
Cosmic Background Radiation and Decoupling .
-
CPT Symmetries
-
Dark Matter .
-
Friedmann-Robertson-Walker Equations .
-
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
-
Vacuum Energy .

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

-
Basis One-forms .
-
Catalog of Spacetimes .
-
Curvature and Parallel Transport
-
Einstein's Field Equations
-
Geodesics
-
Gravitational Waves
-
Hyperbolic Motion and Rindler Coordinates .
-
Quantum Gravity
-
Ricci Decomposition .
-
Ricci Flow .
-
Stress-Energy-Momentum Tensor .
-
Tensors
-
The Area Metric
-
The Dirac Equation in Curved Spacetime .
-
The Equivalence Principal
-
The Essential Mathematics of General Relativity
-
The Induced Metric
-
The Light Cone .
-
The Metric Tensor .
-
The Principle of Least Action in Relativity .
-
Vierbein (Frame) Fields

Group Theory

-
Basic Group Theory .
-
Basic Representation Theory .
-
Building Groups From Other Groups .
-
Sets, Groups, Modules, Rings and Vector Spaces
-
Symmetric Groups .
-
The Integers Modulo n Under + and x .

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
-
Binomial Theorem (Pascal's Triangle)
-
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
-
Similar Matrices and Diagonalization .
-
Spherical Trigonometry
-
Stirling's Approximation
-
Sum and Differences of Squares and Cubes
-
Symbolic Logic
-
Tangent and Normal Line
-
Taylor and Maclaurin Series .
-
The Essential Mathematics of Lie Groups
-
The Limit Definition of the Exponential Function
-
Tic-Tac-Toe Factoring
-
Trapezoidal Rule
-
Unit Vectors
-
Volume Integrals

Microeconomics

-
Marginal Revenue and Cost

Nuclear Physics

-
-
Radioactive Decay

Particle Physics

-
Feynman Diagrams and Loops
-
Field Dimensions
-
Helicity, Chirality and Weyl Spinors .
-
Klein-Gordon and Dirac Equations .
-
Regularization and Renormalization
-
Scattering - Mandelstam Variables
-
Spin 1 Eigenvectors .

Probability and Statistics

-
Box and Whisker Plots
-
Buffon's Needle .
-
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

-
Density Operators and Mixed States .
-
Entangled States .
-
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

-
Bohr Atom
-
Clebsch-Gordan Coefficients .
-
Commutators
-
Dyson Series
-
Electron Orbital Angular Momentum and Spin
-
Heisenberg Uncertainty Principle
-
Ladder Operators .
-
Multi Electron Wavefunctions .
-
Pauli Spin Matrices
-
Photoelectric Effect .
-
Position and Momentum States .
-
Probability Current
-
Schrodinger Equation for Hydrogen Atom .
-
Schrodinger Wave Equation
-
Spin 1/2 Eigenvectors
-
The Differential Operator
-
The Essential Mathematics of Quantum Mechanics
-
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 (Faraday) Tensor .
-
Energy and Momentum in Special Relativity, E = mc2 .
-
Invariance of the Velocity of Light .
-
Lorentz Invariance .
-
Lorentz Transform .
-
Lorentz Transformation of the EM Field .
-
Newton versus Einstein
-
Spinors - Part 1 .
-
Spinors - Part 2 .
-
The Continuity Equation .
-
The Lorentz Group .

Statistical Mechanics

-
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

-
Bardeen–Cooper–Schrieffer Theory
-
BCS Theory
-
Cooper Pairs
-
Introduction to Superconductivity .
-
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
Last modified: February 13, 2022 ✓

Blackbody Radiation ------------------- Blackbody radiation is a type of electromagnetic radiation emitted by a blackbody (a hypothetical body that completely absorbs all radiant energy falling upon it, reaches some equilibrium temperature, and then reemits that energy as quickly as it absorbs it). It can either be within or surround a body in thermodynamic equilibrium with its environment. It can also be emitted by an opaque and non-reflective 'black' body held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body. The radiated energy can be considered to be produced by standing waves or resonant modes of the cavity which is radiating. Consider electromagnetic radiation inside a cavity with sides, L. The standing waves in the cavity have to satisfy the wave equation: ∇2E = (1/c2)∂2/∂t2. With solution: E = E0sin(2πx/λ)sin(2πy/λ)sin(2πz/λ)sin(2πct/λ) This can be written as: E = E0sin(kxx)sin(kyy)sin(kzz)sin(2πct/λ) where k = 2π/λ The solution must give zero amplitude at the walls, since a non-zero value would dissipate energy and violate the condition of equilibrium. This condition can be met by: L = nλ/2 ∴ λ = 2L/n Now k = 2π/λ = nπ/L In 3D we can write: k2 = kx2 + ky2 + kz2 = (nx2 + ny2 + nz22/L2   = n2π2/L2 The number of states, N, in n space occupies 1/8 of a sphere. The volume of a shell of radius n and thickness dn is: N(n)dn = (1/8)4πn2dn Now n2 = k2L22 so n = kL/π and dn = Ldk/π N(k)dk = (1/8)(4πk2L22)(L/π)dk = (1/2π2)L3k2dk N(k) = (1/2π2)L3∫k2dk = (1/6π2)L3k3 ω = ck gives: N(ω) = (1/6π2)(L3/c33 ω = 2πν gives: N(ν) = (4π/3)(L3/c33 k = 2π/λ gives: N(λ) = 4πL3/3λ3 Now there are 2 polarizations allowed for the photon. Therefore, the constant needs to be multiplied by 2: N(ω) = (1/3π2)(L3/c33 N(ν) = (8π/3)(L3/c33 N(λ) = 8πL3/3λ3 This is the total number of modes. To get the distribution by ω, ν or λ it is necessary to take the derivative of the number of modes with respect to ω, ν or λ (i.e., the rates of change). Therefore: dN(ω)/dω = (1/π2)(L3/c32 dN(ν)/dν = (8πL3/c32 check: ω = 2πν and dω = 2πdν ∴ (1/2π)dN(ν)/dν = (1/π2)(L3/c3)4π2ν2 ∴ dN(ν)/dν = (8πL3/c32 dN(λ)/dλ = -8πL34 check: ν = c/λ and dν = -c/λ2dλ ∴ (-λ2/c)dN(λ)/dλ = (8π)(L3/c3)c22 ∴ dN(dλ)/dλ = -8πL34 In the latter case, the negative sign shows that the number of modes decreases with increasing wavelength. Equipartion of Energy --------------------- The theorem of equipartition of energy states that each mode of radiation associated with a classical harmonic oscillator has an energy equal to KBT. Therefore, the energy of the radiation per unit ω, ν or λ per unit volume (≡ energy density per unit ω, ν or λ) is: Eω = (ω22c3)(KBT) or, Eν = (8πν2/c3)(KBT) or, Eλ = (8π/λ4)(KBT)) Dimensional check: LHS: [E/ωV] = [mL2/t2/(1/t)L3] = [m/Lt] RHS: [(mL2/t2)(1/t2)/L3/t3] = [m/Lt] This is the classical RAYLEIGH-JEANS law that says the energy is proportional to the square of the frequency. Since there was no reason to believe that their existed a maximum frequency inside the cavity, the energy would increase towards infinity as the frequency is increased. However, experimentally this did not happen. This was referred to as the ULTRAVIOLET CATASTROPHE. It wasn't until the advent of Quantum Mechanics that this paradox was solved. Planck assumed that the sources of radiation are atoms in a state of oscillation and that the vibrational energy of each oscillator may have any of a series of discrete values but never any value between. Planck further assumed that when an oscillator changes from a higher state of energy to a lower state of energy a discrete quantum of radiation given by E = hν is emitted. Planck was able to calculate the value of h from experimental data. His result, 6.55 x 10-34 J/s, is within 1.2% of the currently accepted value. He was also able to make the first determination of the Boltzmann constant KB from the same data and theory. If we replace the classical oscillator with the quantum oscillator we get: KBT -> hν/(exp(hν/KBT) - 1) Which results in: Eω = (ω22c3)[hω/((exp(hω/KBT) - 1)] or, Eν = (8πν2/c3)[hν/((exp(hν/KBT) - 1)] or, Eλ = (8π/λ4)[hc/λ((exp(hc/KBλT) - 1)] This is PLANCK'S RADIATION LAW. Note: For small ν, exp(x) - 1 = x so the formulae reduces to the classical case. A comparison between the Rayleigh-Jeans law and Planck's law is shown below. Stefan-Boltzmann Law -------------------- The Stefan-Boltzmann law gives the total energy radiated per unit surface area of a black body across all wavelengths per unit time. It is the radiative power per unit area. To find the radiated power per unit area from a surface it is necessary to multiply the energy density by c/4. This comes about because in thermal equilibrium the inward energy flow equals the outward flow giving a factor of 1/2 for the radiated power outward. Then it is required to average over all angles, which gives another factor of 1/2. Therefore, the radiated power per unit area as a function of wavelength is: Pλ = (c/4)(8π/λ4)[hc/λ((exp(hc/KBλT) - 1)] = (2πhc25)[1/((exp(hc/KBλT) - 1)] If we make the substitution x = hc/KBT and dx = -(hc/λ2KBT)dλ, we can write: Pλ = 2π(KBT)4/(h3c2)∫x3/(exp(x) - 1)dx This is a standard integral equal to π4/15. Thus, we get: Pλ = (2π5KB4/15h3c2)T4 = σT4 To get the total power in Watts it is necessary to multiply the above equation by the area, A. For hot objects other than ideal radiators, the law is expressed in the form: Pλ = eAσT4 where e is the emissivity of the object (e = 1 for an ideal radiator). If the object is radiating/absorbing energy to its surroundings at temperature Tenvironment, the net radiation loss/gain rate takes the form: Pλ = eAσ(Tobject4 - Tenvironment4) If Tobject > Tenvironment the object is radiating. If Tobject < Tenvironment the object is absorbing. Wien's Displacement Law ----------------------- Return to the Planck radiation law: Eλ = (8πL3/3λ4)hν/((exp(hc/KBTλ) - 1) The maximum value of this function can be found by differentiating w.r.t. λ, setting the result = 0 and solving for λ. This gives: λT = (hc/KB)/5[1 - exp(-hc/KBλT)] Which must be solved numerically to give: λpeakT = 2.898 x 10-3 meter Kelvin This is the WIEN DISPLACEMENT LAW. It says that as the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. When the maximum is evaluated from the Planck radiation formula, the product of the peak wavelength and the temperature is found to be a constant. This is useful for the determining the temperatures of objects whose temperature is far above that of their surroundings, such as hot radiant stars, by looking at their spectra.