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Astronomy

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Astronomical Distance Units .

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Celestial Navigation .

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Chemistry

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Balancing Chemical Equations

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The Periodic Table .

Classical Physics

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Archimedes Principle

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Comparison Between Gravitation and Electrostatics

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Compton Effect .

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General Relativity

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Accelerated Reference Frames - Rindler Coordinates

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Heron's Formula

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Index Notation (Tensors and Matrices)

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Integration By Parts

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Introduction to Conformal Field Theory .

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Inverse of a Function

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Spherical Trigonometry

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Symmetric Groups

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Tangent and Normal Line

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Taylor and Maclaurin Series .

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The Integers Modulo n Under + and x

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Vector Calculus

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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Field Dimensions

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Helicity and Chirality

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Scattering - Mandelstam Variables

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Spin 1 Eigenvectors .

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The Vacuum Catastrophe

Probability and Statistics

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Chebyshev's Theorem

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Factor Analysis

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Hypothesis Testing

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Linear Regression

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Permutations and Combinations

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How this site works ...

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Quantum Computing

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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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Clebsch-Gordan Coefficients .

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Electron Orbital Angular Momentum and Spin

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Entangled States

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Heisenberg Uncertainty Principle

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Ladder Operators .

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Multi Electron Wavefunctions

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Pauli Exclusion Principle

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Pauli Spin Matrices

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Photoelectric Effect

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Position and Momentum States

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Probability Current

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Schrodinger Equation for Hydrogen Atom

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Schrodinger Wave Equation

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Schrodinger Wave Equation (continued)

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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 Quantum Harmonic Oscillator .

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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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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 P-N Junction

Special Relativity

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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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Lorentz Transformation of the EM Field

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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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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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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

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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Last modified: January 26, 2018


Schrodinger Wave Equation
-------------------------
Assume electron is a travelling wave (free particle) defined by:

ψ = Aexpi(kx-ωt))

Spatial derivative:

∂ψ/∂x = ikψ = (ip/h)ψ 

Multiply both sides by ih to get:

-ih∂ψ/∂x = pψ  

∴ (h²/2m)∂²ψ/∂x² = (p²/2m)ψ 

                = Eψ   ... A.
    
This is the TIME INDEPENDENT form.
                                        
Time derivative:

In addition to its role in determining system energies, the Hamiltonian 
operator also generates the time evolution of the wavefunction. 

∂ψ/∂t = -iωψ = (-iE/h)ψ since E = hω

Multiply both sides by ih to get:

∴ ih∂ψ/∂t = Eψ  ... B.

Equating A. with B. we get:
                                                      
-(h²/2m)∂²ψ/∂x² = ih∂ψ/∂t 

This is the TIME DEPENDENT form.

The time-independent Schrodinger equation is the equation describing 
stationary states (atomic or molecular orbitals).  It is only used when 
the Hamiltonian itself is not dependent on time.  Fortunately, the majority 
of interesting problems in quantum mechanics involve stationary states 
with definite total energy and so the time dependent form is not required.

Equation in the presence of a potential:

Time Independent form:

-(h²/2m)∂²ψ(x)/∂x² + U(x)ψ(x) = Eψ(x)

Time Dependent form:

-(h²/2m)∂²ψ(x,t)/∂x² + U(x)ψ(x,t) = ih∂ψ(x,t)/∂t