Quantum Mechanics for Electrical Engineers

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.

• Preface xiiiAcknowledgments xvAbout the Author xvii1. Introduction 11.1 Why Quantum Mechanics? 11.1.1 Photoelectric Effect 11.1.2 Wave–Particle Duality 21.1.3 Energy Equations 31.1.4 The Schrödinger Equation 51.2 Simulation of the One-Dimensional Time-Dependent Schrödinger Equation 71.2.1 Propagation of a Particle in Free Space 81.2.2 Propagation of a Particle Interacting with a Potential 111.3 Physical Parameters: The Observables 141.4 The Potential V(x) 171.4.1 The Conduction Band of a Semiconductor 171.4.2 A Particle in an Electric Field 171.5 Propagating through Potential Barriers 201.6 Summary 23Exercises 24References 252. Stationary States 272.1 The Infinite Well 282.1.1 Eigenstates and Eigenenergies 302.1.2 Quantization 332.2 Eigenfunction Decomposition 342.3 Periodic Boundary Conditions 382.4 Eigenfunctions for Arbitrarily Shaped Potentials 392.5 Coupled Wells 412.6 Bra-ket Notation 442.7 Summary 47Exercises 47References 493. Fourier Theory in Quantum Mechanics 513.1 The Fourier Transform 513.2 Fourier Analysis and Available States 553.3 Uncertainty 593.4 Transmission via FFT 623.5 Summary 66Exercises 67References 694. Matrix Algebra in Quantum Mechanics 714.1 Vector and Matrix Representation 714.1.1 State Variables as Vectors 714.1.2 Operators as Matrices 734.2 Matrix Representation of the Hamiltonian 764.2.1 Finding the Eigenvalues and Eigenvectors of a Matrix 774.2.2 A Well with Periodic Boundary Conditions 774.2.3 The Harmonic Oscillator 804.3 The Eigenspace Representation 814.4 Formalism 834.4.1 Hermitian Operators 834.4.2 Function Spaces 84Appendix: Review of Matrix Algebra 85Exercises 88References 905. A Brief Introduction to Statistical Mechanics 915.1 Density of States 915.1.1 One-Dimensional Density of States 925.1.2 Two-Dimensional Density of States 945.1.3 Three-Dimensional Density of States 965.1.4 The Density of States in the Conduction Band of a Semiconductor 975.2 Probability Distributions 985.2.1 Fermions versus Classical Particles 985.2.2 Probability Distributions as a Function of Energy 995.2.3 Distribution of Fermion Balls 1015.2.4 Particles in the One-Dimensional Infinite Well 1055.2.5 Boltzmann Approximation 1065.3 The Equilibrium Distribution of Electrons and Holes 1075.4 The Electron Density and the Density Matrix 1105.4.1 The Density Matrix 111Exercises 113References 1146. Bands and Subbands 1156.1 Bands in Semiconductors 1156.2 The Effective Mass 1186.3 Modes (Subbands) in Quantum Structures 123Exercises 128References 1297. The Schrödinger Equation for Spin-1/2 Fermions 1317.1 Spin in Fermions 1317.1.1 Spinors in Three Dimensions 1327.1.2 The Pauli Spin Matrices 1357.1.3 Simulation of Spin 1367.2 An Electron in a Magnetic Field 1427.3 A Charged Particle Moving in Combined E and B Fields 1467.4 The Hartree–Fock Approximation 1487.4.1 The Hartree Term 1487.4.2 The Fock Term 153Exercises 155References 1578. The Green’s Function Formulation 1598.1 Introduction 1608.2 The Density Matrix and the Spectral Matrix 1618.3 The Matrix Version of the Green’s Function 1648.3.1 Eigenfunction Representation of Green’s Function 1658.3.2 Real Space Representation of Green’s Function 1678.4 The Self-Energy Matrix 1698.4.1 An Electric Field across the Channel 1748.4.2 A Short Discussion on Contacts 175Exercises 176References 1769. Transmission 1779.1 The Single-Energy Channel 1779.2 Current Flow 1799.3 The Transmission Matrix 1819.3.1 Flow into the Channel 1839.3.2 Flow out of the Channel 1849.3.3 Transmission 1859.3.4 Determining Current Flow 1869.4 Conductance 1899.5 Büttiker Probes 1919.6 A Simulation Example 194Exercises 196References 19710. Approximation Methods 19910.1 The Variational Method 19910.2 Nondegenerate Perturbation Theory 20210.2.1 First-Order Corrections 20310.2.2 Second-Order Corrections 20610.3 Degenerate Perturbation Theory 20610.4 Time-Dependent Perturbation Theory 20910.4.1 An Electric Field Added to an Infinite Well 21210.4.2 Sinusoidal Perturbations 21310.4.3 Absorption, Emission, and Stimulated Emission 21510.4.4 Calculation of Sinusoidal Perturbations Using Fourier Theory 21610.4.5 Fermi’s Golden Rule 221Exercises 223References 22511. The Harmonic Oscillator 22711.1 The Harmonic Oscillator in One Dimension 22711.1.1 Illustration of the Harmonic Oscillator Eigenfunctions 23211.1.2 Compatible Observables 23311.2 The Coherent State of the Harmonic Oscillator 23311.2.1 The Superposition of Two Eigentates in an Infinite Well 23411.2.2 The Superposition of Four Eigenstates in a Harmonic Oscillator 23511.2.3 The Coherent State 23611.3 The Two-Dimensional Harmonic Oscillator 23811.3.1 The Simulation of a Quantum Dot 238Exercises 244References 24412. Finding Eigenfunctions Using Time-Domain Simulation 24512.1 Finding the Eigenenergies and Eigenfunctions in One Dimension 24512.1.1 Finding the Eigenfunctions 24812.2 Finding the Eigenfunctions of Two-Dimensional Structures 24912.2.1 Finding the Eigenfunctions in an Irregular Structure 25212.3 Finding a Complete Set of Eigenfunctions 257Exercises 259References 259Appendix A. Important Constants and Units 261Appendix B. Fourier Analysis and the Fast Fourier Transform (FFT) 265B.1 The Structure of the FFT 265B.2 Windowing 267B.3 FFT of the State Variable 270Exercises 271References 271Appendix C. An Introduction to the Green’s Function Method 273C.1 A One-Dimensional Electromagnetic Cavity 275Exercises 279References 279Appendix D. Listings of the Programs Used in this Book 281D.1 Chapter 1 281D.2 Chapter 2 284D.3 Chapter 3 295D.4 Chapter 4 309D.5 Chapter 5 312D.6 Chapter 6 314D.7 Chapter 7 323D.8 Chapter 8 336D.9 Chapter 9 345D.10 Chapter 10 356D.11 Chapter 11 378D.12 Chapter 12 395D.13 Appendix B 415Index 419