Introduction to Quantum Mechanics and its Applications

Physics 438A, 4 Units

Spring 2005

2:00pm - 3:50pm, TTH, WPH B26

 

Instructor: Tony Levi Office Hours:
Information on research group: http://www.usc.edu/alevi TTH  8:00 a.m. - 9:30 a. m.
Office: KAP  132 or by appointment
Phone: 213.740.7318 Course outline:
E-mail: alevi@usc.edu  Introduction to Quantum Mechanics and its Applications (This document and all handouts are in PDF format.)

Web site:

http://physics.usc.edu/Classes/

 

   
Grading: Final Exam:
Midterm 35% 2:00 pm - 4:00 pm
Homework 20% May 3, 2005
Final Exam  45% WPH B26
     
Required Text:  
Applied Quantum Mechanics Last day of classes Friday, April 28
A.F.J. Levi  
Cambridge University Press (2003)  
ISBN 052181765X
Additional problems  
   
Secondary Text:  
A Modern Approach to Quantum Mechanics  
John S. Townsend  
Mcgraw-Hill College (1992)  
ISBN: 0070651191  

 

 

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 Maxwellís 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 Maxwellís 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 SCHR÷DINGER 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 SchrŲdinger 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 SCHR÷DINGER 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

                        Blochís 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 Ė 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

FERMIíS 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

                        Fermiís 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