Introduction
to Quantum Mechanics and its Applications
Physics
438A,
4 Units
Spring 2005
2:00pm -
3:50pm, TTH, WPH B26
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
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
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
Heterostructure bipolar transistor with resonant tunnel barrier
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
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
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
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
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
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
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
