Quantum Mechanics for Electrical Engineers

Explains quantum mechanics in language that electrical engineersunderstand

As semiconductor devices become smaller and smaller, classicalphysics alone cannot fully explain their behavior. Instead,electrical engineers need to understand the principles of quantummechanics in order to successfully design and work with today′ssemiconductors.

Written by an electrical engineering professor for students andprofessionals in electrical engineering, Quantum Mechanics forElectrical Engineers focuses on those topics in quantummechanics that are essential for modern semiconductor theory.

This book begins with an introduction to the field, explainingwhy classical physics fails when dealing with very small particlesand small dimensions. Next, the author presents a variety of topicsin quantum mechanics, including:

The Schrödinger equation

Fourier theory in quantum mechanics

Matrix theory in quantum mechanics

An introduction to statistical mechanics

Transport in semiconductors

Because this book is written for electrical engineers, theexplanations of quantum mechanics are rooted in mathematics such asFourier theory and matrix theory that are familiar to allelectrical engineers. Beginning with the first chapter, the authoremploys simple MATLAB computer programs to illustrate keyprinciples. These computer programs can be easily copied and usedby readers to become more familiar with the material. They can alsobe used to perform the exercises at the end of each chapter.

Quantum Mechanics for Electrical Engineers is recommendedfor upper–level undergraduates and graduate students as well asprofessional electrical engineers who want to understand thesemiconductors of today and the future.

Preface xiii Acknowledgments xv About the Author xvii 1. Introduction 1 1.1 Why Quantum Mechanics?, 1 1.1.1 Photoelectric Effect, 1 1.1.2 Wave Particle Duality, 2 1.1.3 Energy Equations, 3 1.1.4 The Schrödinger Equation, 5 1.2 Simulation of the One–Dimensional, Time–DependentSchrödinger Equation, 7 1.2.1 Propagation of a Particle in Free Space, 8 1.2.2 Propagation of a Particle Interacting with a Potential,11 1.3 Physical Parameters: The Observables, 14 1.4 The Potential V ( x ), 17 1.4.1 The Conduction Band of a Semiconductor, 17 1.4.2 A Particle in an Electric Field, 17 1.5 Propagating through Potential Barriers, 20 1.6 Summary, 23 Exercises, 24 References, 25 2. Stationary States 27 2.1 The Infi nite Well, 28 2.1.1 Eigenstates and Eigenenergies, 30 2.1.2 Quantization, 33 2.2 Eigenfunction Decomposition, 34 2.3 Periodic Boundary Conditions, 38 2.4 Eigenfunctions for Arbitrarily Shaped Potentials, 39 2.5 Coupled Wells, 41 2.6 Bra–ket Notation, 44 2.7 Summary, 47 Exercises, 47 References, 49 3. Fourier Theory in Quantum Mechanics 51 3.1 The Fourier Transform, 51 3.2 Fourier Analysis and Available States, 55 3.3 Uncertainty, 59 3.4 Transmission via FFT, 62 3.5 Summary, 66 Exercises, 67 References, 69 4. Matrix Algebra in Quantum Mechanics 71 4.1 Vector and Matrix Representation, 71 4.1.1 State Variables as Vectors, 71 4.1.2 Operators as Matrices, 73 4.2 Matrix Representation of the Hamiltonian, 76 4.2.1 Finding the Eigenvalues and Eigenvectors of a Matrix,77 4.2.2 A Well with Periodic Boundary Conditions, 77 4.2.3 The Harmonic Oscillator, 80 4.3 The Eigenspace Representation, 81 4.4 Formalism, 83 4.4.1 Hermitian Operators, 83 4.4.2 Function Spaces, 84 Appendix: Review of Matrix Algebra, 85 Exercises, 88 References, 90 5. A Brief Introduction to Statistical Mechanics 91 5.1 Density of States, 91 5.1.1 One–Dimensional Density of States, 92 5.1.2 Two–Dimensional Density of States, 94 5.1.3 Three–Dimensional Density of States, 96 5.1.4 The Density of States in the Conduction Band of aSemiconductor, 97 5.2 Probability Distributions, 98 5.2.1 Fermions versus Classical Particles, 98 5.2.2 Probability Distributions as a Function of Energy, 99 5.2.3 Distribution of Fermion Balls, 101 5.2.4 Particles in the One–Dimensional Infi nite Well, 105 5.2.5 Boltzmann Approximation, 106 5.3 The Equilibrium Distribution of Electrons and Holes, 107 5.4 The Electron Density and the Density Matrix, 110 5.4.1 The Density Matrix, 111 Exercises, 113 References, 114 6. Bands and Subbands 115 6.1 Bands in Semiconductors, 115 6.2 The Effective Mass, 118 6.3 Modes (Subbands) in Quantum Structures, 123 Exercises, 128 References, 129 7. The Schrödinger Equation for Spin–1/2 Fermions131 7.1 Spin in Fermions, 131 7.1.1 Spinors in Three Dimensions, 132 7.1.2 The Pauli Spin Matrices, 135 7.1.3 Simulation of Spin, 136 7.2 An Electron in a Magnetic Field, 142 7.3 A Charged Particle Moving in Combined E and B Fields, 146 7.4 The Hartree Fock Approximation, 148 7.4.1 The Hartree Term, 148 7.4.2 The Fock Term, 153 Exercises, 155 References, 157 8. The Green s Function Formulation 159 8.1 Introduction, 160 8.2 The Density Matrix and the Spectral Matrix, 161 8.3 The Matrix Version of the Green s Function, 164 8.3.1 Eigenfunction Representation of Green s Function,165 8.3.2 Real Space Representation of Green s Function,167 8.4 The Self–Energy Matrix, 169 8.4.1 An Electric Field across the Channel, 174 8.4.2 A Short Discussion on Contacts, 175 Exercises, 176 References, 176 9. Transmission 177 9.1 The Single–Energy Channel, 177 9.2 Current Flow, 179 9.3 The Transmission Matrix, 181 9.3.1 Flow into the Channel, 183 9.3.2 Flow out of the Channel, 184 9.3.3 Transmission, 185 9.3.4 Determining Current Flow, 186 9.4 Conductance, 189 9.5 Büttiker Probes, 191 9.6 A Simulation Example, 194 Exercises, 196 References, 197 10. Approximation Methods 199 10.1 The Variational Method, 199 10.2 Nondegenerate Perturbation Theory, 202 10.2.1 First–Order Corrections, 203 10.2.2 Second–Order Corrections, 206 10.3 Degenerate Perturbation Theory, 206 10.4 Time–Dependent Perturbation Theory, 209 10.4.1 An Electric Field Added to an Infinite Well, 212 10.4.2 Sinusoidal Perturbations, 213 10.4.3 Absorption, Emission, and Stimulated Emission, 215 10.4.4 Calculation of Sinusoidal Perturbations Using FourierTheory, 216 10.4.5 Fermi s Golden Rule, 221 Exercises, 223 References, 225 11. The Harmonic Oscillator 227 11.1 The Harmonic Oscillator in One Dimension, 227 11.1.1 Illustration of the Harmonic Oscillator Eigenfunctions,232 11.1.2 Compatible Observables, 233 11.2 The Coherent State of the Harmonic Oscillator, 233 11.2.1 The Superposition of Two Eigentates in an Infinite Well,234 11.2.2 The Superposition of Four Eigenstates in a HarmonicOscillator, 235 11.2.3 The Coherent State, 236 11.3 The Two–Dimensional Harmonic Oscillator, 238 11.3.1 The Simulation of a Quantum Dot, 238 Exercises, 244 References, 244 12. Finding Eigenfunctions Using Time–Domain Simulation245 12.1 Finding the Eigenenergies and Eigenfunctions in OneDimension, 245 12.1.1 Finding the Eigenfunctions, 248 12.2 Finding the Eigenfunctions of Two–Dimensional Structures,249 12.2.1 Finding the Eigenfunctions in an Irregular Structure,252 12.3 Finding a Complete Set of Eigenfunctions, 257 Exercises, 259 References, 259 Appendix A. Important Constants and Units 261 Appendix B. Fourier Analysis and the Fast Fourier Transform(FFT) 265 B.1 The Structure of the FFT, 265 B.2 Windowing, 267 B.3 FFT of the State Variable, 270 Exercises, 271 References, 271 Appendix C. An Introduction to the Green s FunctionMethod 273 C.1 A One–Dimensional Electromagnetic Cavity, 275 Exercises, 279 References, 279 Appendix D. Listings of the Programs Used in this Book281 D.1 Chapter 1, 281 D.2 Chapter 2, 284 D.3 Chapter 3, 295 D.4 Chapter 4, 309 D.5 Chapter 5, 312 D.6 Chapter 6, 314 D.7 Chapter 7, 323 D.8 Chapter 8, 336 D.9 Chapter 9, 345 D.10 Chapter 10, 356 D.11 Chapter 11, 378 D.12 Chapter 12, 395 D.13 Appendix B, 415 Index 419 MATLAB Coes are downloadable from http://booksupport.wiley.com

DENNIS M. SULLIVAN is Professor of Electrical and Computer Engineering at the University of Idaho as well as an award–winning author and researcher. In 1997, Dr. Sullivan′s paper "Z Transform Theory and FDTD Method" won the IEEE Antennas and Propagation Society′s R. P. W. King Award for the Best Paper by a Young Investigator. He is the author of Electromagnetic Simulation Using the FDTD Method.