Prerequisites

Mathematics:
A basic working knowledge of differential calculus, Fourier analysis, linear algebra, statistics and geometry.
Computer skills:
An ability to program numerical algorithms in C, MATLAB, FORTRAN or similar language and display results in graphical form.
Physics background:
Should include a basic understanding of Newtonian mechanics, waves and Maxwell’s equations.

Introduction: Lectures 1 – 4
Lecture 1
CLASSICAL MECHANICS
Introduction to force, potential, and the Hamiltonian
The one-dimensional simple harmonic oscillator
Harmonic oscillation of a diatomic molecule
Lattice dynamics of the monoatomic and diatomic linear chain

Lecture 2
CLASSICAL ELECTROMAGNETISM
Electrostatic force and potential between charges
The parallel plate capacitor
The Coulomb blockade
Electrodynamics and Maxwell’s equations
Light propagation in a dielectric medium
Power and momentum in an electromagnetic wave
Choosing a scalar or vector potential
Dipole radiation

Lecture 3
TOWARDS QUANTUM MECHANICS
Diffraction, interference, and correlation functions for light
Black-body radiation and evidence for quantization of light
Photoelectric effect and the photon particle
Secure quantum communication
The link between quantization of photons and quantization of other particles
Diffraction and interference of electrons
When is a particle a wave?

Lecture 4
THE SCHRÖDINGER WAVE EQUATION
The wave function description of an electron of mass m0 in free-space
The electron wave packet and dispersion
The Bohr model of the hydrogen atom
Calculation of the average radius of an electron orbit in hydrogen
Calculation of energy difference between electron orbits in hydrogen
Periodic table of elements
Crystal structure
Three types of solid classified according to atomic arrangement
Two-dimensional square lattice
Cubic lattices in three-dimensions
Electronic properties of semiconductor crystals
The semiconductor heterostructure

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Using the Schrödinger wave equation: Lectures 5 – 6
Lecture 5
INTRODUCTION
The effect of discontinuities in the wave function and its derivative
WAVE FUNCTION NORMALIZATION AND COMPLETENESS
INVERSION SYMMETRY IN THE POTENTIAL
Particle in a one-dimensional square potential well with infinite barrier energy
NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION
CURRENT FLOW
Current flow in a one-dimensional infinite square potential well
Current flow due to a traveling wave
DEGENERACY IS A CONSEQUENCE OF SYMMETRY
Bound states in three-dimensions and degeneracy of eigenvalues
BOUND STATES OF A SYMMETRIC SQUARE POTENTIAL WELL
Symmetric rectangular potential well with finite barrier energy

Lecture 6
TRANSMISSION AND REFLECTION OF UNBOUND STATES
Scattering from a potential step when effective electron mass changes
Probability current density for scattering at a step
Impedance matching for unity transmission
PARTICLE TUNNELING
Electron tunneling limit to reduction in size of CMOS transistors
THE NONEQUILIBRIUM ELECTRON TRANSISTOR

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Scattering in one-dimension: The propagation method: Lectures 7 – 11
Lecture 7
THE PROPAGATION MATRIX METHOD
Writing a computer program for the propagation method
TIME REVERSAL SYMMETRY
CURRENT CONSERVATION AND THE PROPAGATION MATRIX

Lecture 8
THE RECTANGULAR POTENTIAL BARRIER
Tunneling
RESONANT TUNNELING
Localization threshold
Multiple potential barriers
THE POTENIAL BARRIER IN THE d-FUNCTION LIMIT

Lecture 9
ENERGY BANDS IN PERIODIC POTENTIALS: THE KRONIG-PENNY POTENTIAL
Bloch’s theorem
Propagation matrix in a periodic potential

Lecture 10
THE TIGHT BINDING MODEL FOR ELECTRONIC BANDSTRUCTURE
Nearest neighbor and long-range interactions
Crystal momentum and effective electron mass
USE OF THE PROPAGATION MATRIX TO SOLVE OTHER PROBLEMS IN ENGINEERING
THE WKB APPROXIMATION
Tunneling

Lecture 11
RESEARCH LECTURE #2
Light coupling to a photonic crystal super-prism

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Related mathematics: Lecture 12 – 13
Lecture 12
ONE PARTICLE WAVE FUNCTION SPACE
PROPERTIES OF LINEAR OPERATORS
DIRAC NOTATION
MEASUREMENT OF REAL NUMBERS
COMMUTATING OPERATORS
THE GENERALIZED UNCERTAINTY RELATION

Lecture 13
DENSITY OF STATES
Plane wave density of states
Quantum well and quantum dot
Numerically evaluating density of states from a dispersion relation
Quantum conductance
Density of photon states

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The harmonic oscillator: Lectures 14 – 15
Lecture 14
THE HARMONIC OSCILLATOR POTENTIAL
RAISING AND LOWERING OPERATORS
The ground state
Excited states
HARMONIC OSCILLATOR WAVE FUNCTIONS
Classical turning point
TIME DEPENDENCE
The superposition operator
Measurement of a superposition state

Lecture 15
Time dependence in the Heisenberg representation
Charged particle in harmonic potential subject to constant electric field
ELECTROMAGNETIC FIELDS
Laser light
Quantization of an electrical resonator
Quantization of lattice vibrations
Quantization of mechanical vibrations

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Fermions and Bosons: Lecture 16 – 17
Lecture 16
INTRODUCTION
The symmetry of indistinguishable particles
Slater determinant
Pauli exclusion principle
Fermion creation and annihilation operators – application to tight-binding Hamiltonian

Lecture 17
FERMI-DIRAC DISTRIBUTION FUNCTION
Equilibrium statistics
Writing a computer program to calculate the Fermi-Dirac distribution
BOSE-EINSTIEN DISTRIBUTION FUNCTION

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Time dependent perturbation theory: Lectures 18 – 20
Lecture 18
FIRST-ORDER TIME-DEPENDENT PERTURBATION THEORY
Abrupt change in potential
Time dependent change in potential
CHARGED PARTICLE IN A HARMONIC POTENTIAL
FIRST-ORDER TIME-DEPENDENT PERTURBATION

Lecture 19
FERMI’S GOLDEN RULE
IONIZED IMPURITY ELASTIC SCATTERING RATE IN GaAs
The coulomb potential
Linear screening of the coulomb potential
Correlation effects in position of dopant atoms
Calculating the electron mean free path

Lecture 20
EMISSION OF PHOTONS DUE TO TRANSITIONS BETWEEN ELECTRONIC STATES
Density of optical modes in three dimensions
Light intensity
Background photon energy density at thermal equilibrium
Fermi’s golden rule for stimulated optical transitions
The Einstein A and B coefficients
Occupation factor for photons in thermal equilibrium in a two-level system
Derivation of the relationship between spontaneous emission rate and gain

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Angular momentum and the Hydrogen atom: Lectures 21 – 22
Lecture 21
ANGULAR MOMENTUM
Classical angular momentum
The angular momentum operator
Eigenvalues of the angular momentum operators Lz and L2
Geometrical representation

Lecture 22
SPHERICAL COORDINATES AND SPHERICAL HARMONICS AND THE HYDROGEN ATOM
The rigid rotator
The hydrogen atom

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Time independent perturbation theory: Lectures 23 – 24
Lecture 23
NON-DEGENERATE CASE
Hamiltonian subject to perturbation W
First-order correction
Second order correction
Harmonic oscillator subject to perturbing potential in x, x2 and x3

Lecture 24
DEGENERATE CASE
Secular equation
Two states
Periodic potential
Two- and three-dimensional harmonic oscillator