Quantum Mechanics for Electrical Engineers

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

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–Dependent Schrödinger 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 a Semiconductor, 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 Fermions 131 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 Fourier Theory, 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 Harmonic Oscillator, 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 Simulation 245 12.1 Finding the Eigenenergies and Eigenfunctions in One Dimension, 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 Function Method 273 C.1 A One–Dimensional Electromagnetic Cavity, 275 Exercises, 279 References, 279 Appendix D. Listings of the Programs Used in this Book 281 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