Department
of Electrical Engineering
Engineering Quantum Mechanics
EE 539,
3 Units
Fall 2009
11:00 am -
12:20 pm, TTH, KAP 140
Abstract
Quantum mechanics is the basis for understanding physical phenomena on
the atomic and nano-meter scale. There are numerous applications of quantum
mechanics in biology, chemistry and engineering. Those with significant
economic impact include semiconductor transistors, lasers, quantum optics and
photonics. As technology advances, an increasing number of new electronic and
opto-electronic devices will operate in ways that can only be understood using
quantum mechanics. Over the next twenty years fundamentally quantum devices
such as single-electron memory cells and photonic signal processing systems will
become common-place. The purpose of this course is to cover a few selected
applications and to provide a solid foundation in the tools and methods of
quantum mechanics. The intent is that this understanding will enable insight
and contributions to future, as yet unknown, applications.
Prerequisites
Mathematics:
A basic working knowledge of
differential calculus, 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 an
Maxwell's equations.
Introduction:
Lectures 1 - 3
Lecture
1
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
Lecture
2
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
3
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
Using the Schrödinger
wave equation: Lectures 4 - 6
Lecture
4
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
Matrix solution to the descretized Schrödinger
equation
Nontransmitting boundary conditions
Periodic
boundary conditions
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
Lecture
5
BOUND STATES OF A SYMMETRIC
SQUARE POTENTIAL WELL
Symmetric square potential well with finite barrier energy
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
Lecture
6
PARTICLE
TUNNELING
Electron tunneling limit to reduction in size of CMOS transistors
THE
NONEQUILIBRIUM ELECTRON TRANSISTOR
Scattering in one-dimension:
The propagation method: Lectures 7 - 10
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
Heterostructure bipolar transistor with resonant tunnel barrier
Localization
threshold
Multiple potential barriers
THE POTENTIAL
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 BAND STRUCTURE
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
Related mathematics:
Lecture 11 - 12
Lecture
11
ONE PARTICLE
WAVE FUNCTION SPACE
PROPERTIES
OF LINEAR OPERATORS
Hermitian operators
Commutator algebra
DIRAC NOTATION
MEASUREMENT
OF REAL NUMBERS
Time dependence of expectation values
Uncertainty in expectation value
The
generalized uncertainty relation
THE NO
CLONING THEOREM
Lecture
12
DENSITY OF
STATES
Density of states of particle mass m in 3D, 2D, 1D and 0D
Quantum conductance
Numerically evaluating density of states from a dispersion relation
Density of photon states
The harmonic oscillator:
Lectures 13 - 14
Lecture
13
THE HARMONIC OSCILLATOR POTENTIAL
CREATION AND ANNIHILATION OPERATORS
The
ground state
Excited
states
HARMONIC
OSCILLATOR WAVE FUNCTIONS
Classical turning point
TIME DEPENDENCE
The superposition operator
Measurement of a superposition state
Lecture
14
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
Fermions and Bosons:
Lecture 15 - 16
Lecture
15
INTRODUCTION
The symmetry of indistinguishable particles
Slater determinant
Pauli exclusion principle
Fermion
creation and annihilation operators – application to tight-binding Hamiltonian
Lecture
16
FERMI-DIRAC DISTRIBUTION FUNCTION
Equilibrium statistics
Writing a computer program to calculate the Fermi-Dirac distribution
BOSE-EINSTIEN DISTRIBUTION FUNCTION
Time dependent perturbation
theory and the laser diode: Lectures 17 - 21
Lecture
17
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
18
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
19
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
Lecture 20
THE SEMICONDUCTOR LASER DIODE
Spontaneous and stimulated emission
Optical gain in a semiconductor
Optical gain in the presence of electron scattering
DESIGNING A LASER CAVITY
Resonant
optical cavity
Mirror loss
and photon lifetime
The Fabry-Perot
laser diode
Rate equation
models
Lecture 21
NUMERICAL METHOD
OF SOLVING RATE EQUATIONS
The
Runge-Kutta method
Large-signal
transient response, cavity
formation
NOISE IN LASER DIODE LIGHT EMISSION
Effect of photon and electron number quantization
Time independent perturbation theory: Lectures 22 - 23
Lecture 22
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
23
DEGENERATE CASE
Secular equation
Two states
Perturbation of two-dimensional harmonic oscillator
Perturbation of two-dimensional potential with infinite barrier
Angular momentum and the hydrogenic atom: Lectures 24 - 26
Lecture 24
ANGULAR MOMENTUM
Classical angular momentum
The angular momentum operator
Eigenvalues of the angular momentum operators Lz and L2
Geometrical representation
Lecture 25
SPHERICAL COORDINATES, SPHERICAL HARMONICS AND THE
HYDROGEN ATOM
Spherical coordinates and spherical harmonics
The rigid rotator
Quantization of the hydrogenic atom
Radial and
angular probability density
Lecture
26
ELECTROMAGNETIC RADIATION
No eigenstate radiation
Superposition of eigenstates
Hydrogenic selection rules for dipole radiation
Fine structure
Hybridization
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