Title Details

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

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