Quantum Mechanics for Electrical Engineers E-Book

Dennis M. Sullivan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: IEEE Press Series on Microelectronic Systems
• Sprache: Englisch

Seitenzahl: 368

Veröffentlichungsjahr: 2011

Beschreibung

The main topic of this book is quantum mechanics, as the title indicates. It specifically targets those topics within quantum mechanics that are needed to understand modern semiconductor theory. It begins with the motivation for quantum mechanics and why classical physics fails when dealing with very small particles and small dimensions. Two key features make this book different from others on quantum mechanics, even those usually intended for engineers: First, after a brief introduction, much of the development is through Fourier theory, a topic that is at the heart of most electrical engineering theory. In this manner, the explanation of the quantum mechanics is rooted in the mathematics familiar to every electrical engineer. Secondly, beginning with the first chapter, simple computer programs in MATLAB are used to illustrate the principles. The programs can easily be copied and used by the reader to do the exercises at the end of the chapters or to just become more familiar with the material.

Many of the figures in this book have a title across the top. This title is the name of the MATLAB program that was used to generate that figure. These programs are available to the reader. Appendix D lists all the programs, and they are also downloadable at http://booksupport.wiley.com

Leseprobe

Table of Contents

Cover

Series page

Title page

Copyright page

DEDICATION

PREFACE

THE LAYOUT OF THE BOOK

THE SIMULATION PROGRAMS

ACKNOWLEDGMENTS

ABOUT THE AUTHOR

1 INTRODUCTION

1.1 WHY QUANTUM MECHANICS?

1.2 SIMULATION OF THE ONE-DIMENSIONAL, TIME-DEPENDENT SCHRÖDINGER EQUATION

1.3 PHYSICAL PARAMETERS: THE OBSERVABLES

1.4 THE POTENTIAL V (x)

1.5 PROPAGATING THROUGH POTENTIAL BARRIERS

1.6 SUMMARY

2 STATIONARY STATES

2.1 THE INFINITE WELL

2.2 EIGENFUNCTION DECOMPOSITION

2.3 PERIODIC BOUNDARY CONDITIONS

2.4 EIGENFUNCTIONS FOR ARBITRARILY SHAPED POTENTIALS

2.5 COUPLED WELLS

2.6 BRA-KET NOTATION

2.7 SUMMARY

3 FOURIER THEORY IN QUANTUM MECHANICS

3.1 THE FOURIER TRANSFORM

3.2 FOURIER ANALYSIS AND AVAILABLE STATES

3.3 UNCERTAINTY

3.4 TRANSMISSION VIA FFT

3.5 SUMMARY

4 MATRIX ALGEBRA IN QUANTUM MECHANICS

4.1 VECTOR AND MATRIX REPRESENTATION

4.2 MATRIX REPRESENTATION OF THE HAMILTONIAN

4.3 THE EIGENSPACE REPRESENTATION

4.4 FORMALISM

APPENDIX: REVIEW OF MATRIX ALGEBRA

5 A BRIEF INTRODUCTION TO STATISTICAL MECHANICS

5.1 DENSITY OF STATES

5.2 PROBABILITY DISTRIBUTIONS

5.3 THE EQUILIBRIUM DISTRIBUTION OF ELECTRONS AND HOLES

5.4 THE ELECTRON DENSITY AND THE DENSITY MATRIX

6 BANDS AND SUBBANDS

6.1 BANDS IN SEMICONDUCTORS

6.2 THE EFFECTIVE MASS

6.3 MODES (SUBBANDS) IN QUANTUM STRUCTURES

7 THE SCHRÖDINGER EQUATION FOR SPIN-1/2 FERMIONS

7.1 SPIN IN FERMIONS

7.2 AN ELECTRON IN A MAGNETIC FIELD

7.3 A CHARGED PARTICLE MOVING IN COMBINED E AND B FIELDS

7.4 THE HARTREE–FOCK APPROXIMATION

8 THE GREEN’S FUNCTION FORMULATION

8.1 INTRODUCTION

8.2 THE DENSITY MATRIX AND THE SPECTRAL MATRIX

8.3 THE MATRIX VERSION OF THE GREEN’S FUNCTION

8.4 THE SELF-ENERGY MATRIX

9 TRANSMISSION

9.1 THE SINGLE-ENERGY CHANNEL

9.2 CURRENT FLOW

9.3 THE TRANSMISSION MATRIX

9.4 CONDUCTANCE

9.5 BÜTTIKER PROBES

9.6 A SIMULATION EXAMPLE

10 APPROXIMATION METHODS

10.1 THE VARIATIONAL METHOD

10.2 NONDEGENERATE PERTURBATION THEORY

10.3 DEGENERATE PERTURBATION THEORY

10.4 TIME-DEPENDENT PERTURBATION THEORY

11 THE HARMONIC OSCILLATOR

11.1 THE HARMONIC OSCILLATOR IN ONE DIMENSION

11.2 THE COHERENT STATE OF THE HARMONIC OSCILLATOR

11.3 THE TWO-DIMENSIONAL HARMONIC OSCILLATOR

12 FINDING EIGENFUNCTIONS USING TIME-DOMAIN SIMULATION

12.1 FINDING THE EIGENENERGIES AND EIGENFUNCTIONS IN ONE DIMENSION

12.2 FINDING THE EIGENFUNCTIONS OF TWO-DIMENSIONAL STRUCTURES

12.3 FINDING A COMPLETE SET OF EIGENFUNCTIONS

APPENDIX A: IMPORTANT CONSTANTS AND UNITS

APPENDIX B: FOURIER ANALYSIS AND THE FAST FOURIER TRANSFORM (FFT)

B.1 THE STRUCTURE OF THE FFT

B.2 WINDOWING

B.3 FFT OF THE STATE VARIABLE

APPENDIX C: AN INTRODUCTION TO THE GREEN’S FUNCTION METHOD

C.1 A ONE-DIMENSIONAL ELECTROMAGNETIC CAVITY

APPENDIX D: LISTINGS OF THE PROGRAMS USED IN THIS BOOK

D.1 CHAPTER 1

D.2 CHAPTER 2

D.3 CHAPTER 3

D.4 CHAPTER 4

D.5 CHAPTER 5

D.6 CHAPTER 6

D.7 CHAPTER 7

D.8 CHAPTER 8

D.9 CHAPTER 9

D.10 CHAPTER 10

D.11 CHAPTER 11

D.12 CHAPTER 12

D.13 APPENDIX B

Index

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Technical Reviewers

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Prof. Steven Hughes, Queens University

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Library of Congress Cataloging-in-Publication Data

ISBN: 978-0-470-87409-7

ISBN: 978-1-118-16977-3(epdf)

ISBN: 978-1-118-16979-7(epub)

ISBN: 978-1-118-16980-3(mobi)

To

My Girl

PREFACE

A physics professor once told me that electrical engineers were avoiding learning quantum mechanics as long as possible. The day of reckoning has arrived. Any electrical engineer hoping to work in the field of modern semiconductors will have to understand some quantum mechanics.

Quantum mechanics is not normally part of the electrical engineering curriculum. An electrical engineering student taking quantum mechanics in the physics department may find it to be a discouraging experience. A quantum mechanics class often has subjects such as statistical mechanics, thermodynamics, or advanced mechanics as prerequisites. Furthermore, there is a greater cultural difference between engineers and physicists than one might imagine.

This book grew out of a one-semester class at the University of Idaho titled “Semiconductor Theory,” which is actually a crash course in quantum mechanics for electrical engineers. In it there are brief discussions on statistical mechanics and the topics that are needed for quantum mechanics. Mostly, it centers on quantum mechanics as it applies to transport in semiconductors. It differs from most books in quantum mechanics in two other very important aspects: (1) It makes use of Fourier theory to explain several concepts, because Fourier theory is a central part of electrical engineering. (2) It uses a simulation method called the finite-difference time-domain (FDTD) method to simulate the Schrödinger equation and thereby provides a method of illustrating the behavior of an electron. The simulation method is also used in the exercises. At the same time, many topics that are normally covered in an introductory quantum mechanics text, such as angular momentum, are not covered in this book.

THE LAYOUT OF THE BOOK

Intended primarily for electrical engineers, this book focuses on a study of quantum mechanics that will enable a better understanding of semiconductors. Chapters 1 through 7 are primarily fundamental topics in quantum mechanics. Chapters 8 and 9 deal with the Green’s function formulation for transport in semiconductors and are based on the pioneering work of Supriyo Datta and his colleagues at Purdue University. The Green’s function is a method for calculating transport through a channel. Chapter 10 deals with approximation methods in quantum mechanics. Chapter 11 talks about the harmonic oscillator, which is used to introduce the idea of creation and annihilation operators that are not otherwise used in this book. Chapter 12 describes a simulation method to determine the eigenenergies and eigenstates in complex structures that do not lend themselves to easy analysis.

THE SIMULATION PROGRAMS

Many of the figures in this book have a title across the top. This title is the name of the MATLAB program that was used to generate that figure. These programs are available to the reader. Appendix D lists all the programs, but they can also be obtained from the following Internet site:

http://booksupport.wiley.com.

The reader will find it beneficial to use these programs to duplicate the figures and perhaps explore further. In some cases the programs must be used to complete the exercises at the end of the chapters. Many of the programs are time-domain simulations using the FDTD method, and they illustrate the behavior of an electron in time. Most readers find these programs to be extremely beneficial in acquiring some intuition for quantum mechanics. A request for the solutions manual needs to be emailed to [email protected].

DENNIS M. SULLIVAN

Department of Electrical and Computer Engineering

University of Idaho

ACKNOWLEDGMENTS

I am deeply indebted to Prof. Supriyo Datta of Purdue University for his help, not only in preparing this book, but in developing the class that led to the book. I am very grateful to the following people for their expertise in editing this book: Prof. Richard Ziolkowski from the University of Arizona; Prof. Fred Barlow, Prof. F. Marty Ytreberg, and Paul Wilson from the University of Idaho; Prof. David Citrin from the Georgia Institute of Technology; Prof. Steven Hughes from Queens University; Prof. Enrique Navarro from the University of Valencia; and Dr. Alexey Maslov from Canon U.S.A. I am grateful for the support of my department chairman, Prof. Brian Johnson, while writing this book. Mr. Ray Anderson provided invaluable technical support. I am also very grateful to Ms. Judy LaLonde for her editorial assistance.

D.M.S.

ABOUT THE AUTHOR

Dennis M. Sullivan graduated from Marmion Military Academy in Aurora, Illinois in 1966. He spent the next 3 years in the army, including a year as an artillery forward observer with the 173rd Airborne Brigade in Vietnam. He graduated from the University of Illinois with a bachelor of science degree in electrical engineering in 1973, and received master’s degrees in electrical engineering and computer science from the University of Utah in 1978 and 1980, respectively. He received his Ph.D. degree in electrical engineering from the University of Utah in 1987.

From 1987 to 1993, he was a research engineer with the Department of Radiation Oncology at Stanford University, where he developed a treatment planning system for hyperthermia cancer therapy. Since 1993, he has been on the faculty of electrical and computer engineering at the University of Idaho. His main interests are electromagnetic and quantum simulation. In 1997, his paper “Z Transform Theory and the FDTD Method,” won the R. P. W. King Award from the IEEE Antennas and Propagation Society. In 2001, he received a master’s degree in physics from Washington State University while on sabbatical leave. He is the author of the book Electromagnetic Simulation Using the FDTD Method, also from IEEE Press.