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: November 17, 2019

Relationship Between Position and Momentum States ------------------------------------------------- The |ψ> is an abstract object that represents a quantum state (statevector). By definition the actual wavefunction is the projection of |ψ> onto the bra basis in either position or momentum space. The projection is simply the inner product. Thus, ψ(x) = <x|ψ> ... wave function in position basis. φ(k) = <k|ψ> ... wave function in momentum basis. Momentum in the position basis: pop|p> = p|p> Therefore, pop<x|p> = p<x|p> -ih∂<x|p>/∂x = p<x|p> With solution: <x|p> = (1/√2π)exp(ipx/h) Position in the momentum basis: xop|x> = x|x> Therefore, xop<p|x> = x<p|x> ih∂<p|x>/∂p = x<p|x> With solution: <p|x> ≡ <x|p>* = (1/√2π)exp(-ipx/h) The justification for -ih∂/∂x <-> ih∂/∂p comes from the Fourier transform (described next) that implies that position space and momentum space can be switched in any formula as long as you also take the complex conjugate. Therefore, Momentum in position space: <x|p|ψ> = -ih∂ψ(x)/∂x Position in momentum space: <p|x|ψ> = ih∂ψ(p)/∂p The Fourier Transform ------------------ To recover the Fourier transform we use the "Resolution of the Identity": Σ|ei><ei| = I i or, +∞ ∫dx |ei><ei| = I -∞ The dyadic product is the formal product between a ket and a bra vector. If |ψ> is built using a complete orthonormal basis, ψ> = |ei><ei|ψ> When applied to an operator from the left and right side we get: 1ψ1 = Σ|ei><ei|(ψΣ|ej><ej| i j = ΣΣψij|ei><ej| ij with the matrix elements: ψij = <ei|ψ|ej> We can make use of the dyad in the following manner: ψ(p) = <p|ψ> = <p|x><x|ψ> = <p|x>ψ(x) = (1/√2π)∫exp(-ipx/h)ψ(x)dx and, ψ(x) = <x|ψ> = <x|p><p|ψ> = <x|p>ψ(p) = (1/√2π)∫exp(ipx/h)ψ(p)dp These are Fourier transform. Therefore, the position and momentum wavefunctions are Fourier Transforms of each other. Projection Operator ------------------- The projection operator satisfies the operator equation PP = P. This means that acting twice with a projection operator on a vector gives the same result as acting once. PP = P2 = (|ei><ei|)(|ei><ei|) = |ei> (<ei|ei>) <ei| = |ei> 1 <ei| = |ei><ei| If applied to a vector, it projects the vector onto the state |β> and generates a new vector in parallel to |α> with a magnitude equal to the projection. |α><β|ψ> = <β|ψ>|α> Therefore, P|ψ> = |ei><ei|ψ> = <ei|ψ>|ei> Projects a given wavefunction |ψ> to a wavefunction along |ei>. We can define an operator that projects a vector to a 2 dimensional subspace as: P|ψ> = |ei><ei|ψ> + |ej><ej|ψ> i ≠ j For example, for the spin 1/2 system the unit operator can be written as a sum of 2 terms since the vector space is 2 dimensional. Using the orthonormal basis vectors |↑> and |↓> for up and down spins along the z axis gives: I = |↑><↑| + |↓><↓| In general, we can define an operator that projects a vector to an N dimensional subspace as: P|ψ> = |ei><ei|ψ> + |ej><ej|ψ> + ... + |en><en|ψ> As a matrix P is just the identity matrix. Dirac Delta Function -------------------- The Dirac δ function is the continuous analog of the Kronecker δ for discrete values. Just as the Kronecker Delta usually appears inside a sum, The Dirac Delta usually appears in an integral. Definitions: δ(x) = {∞ x = 0 {0 x ≠ 0 ∫δ(x)dx = 1 f(a) = ∫f(a)δ(x - a) dx ... 1. -∞ As the limit of the Gaussian: δa(x) = lim(1/σ√2π)exp(-x2/2σ2) as a -> 0 σ->0 ψ(x) = (1/√2π)∫exp(ipx/h)ψ(p)dp and, ψ(p) = (1/√2π)∫exp(-ipx/h)ψ(x)dx ψ(x) = (1/√2π)∫dp exp(ipx/h) (1/√2π)∫dx exp(-ipx/h)ψ(x) = (1/2π)∫dp exp(ipx/h)∫dx exp(-ipx/h)ψ(x) = ∫dx'ψ(x') (1/2π)∫dp exp(ip(x - x')h) = ∫dx'ψ(x') δ(x' - x) using 1. So an alternative definition of δ is: δ(x' - x) = (1/2π)∫dp exp(ip(x - x')h) -∞ Orthogonality of Momentum and Position Eigenstates -------------------------------------------------- <x'|x> = <x'|1|x> = <x'|∫dp |p><p||x> = ∫dp <x'|p><p|x> = (1/2πh)∫dp exp(ipx'/h)exp(-ipx/h) = (1/2πh)∫dp exp(ip(x' - x)/h) = δ(x' - x) Likewise, <p'|p> = = δ(p - p')