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 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 Infinite 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 Infinite 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