Quantum Mechanics for Electrical EngineersISBN: 9780470874097
448 pages
January 2012, WileyIEEE Press

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
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 OneDimensional, TimeDependent 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 Braket 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 OneDimensional Density of States, 92
5.1.2 TwoDimensional Density of States, 94
5.1.3 ThreeDimensional 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 OneDimensional 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 Spin1/2 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 SelfEnergy 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 SingleEnergy 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 FirstOrder Corrections, 203
10.2.2 SecondOrder Corrections, 206
10.3 Degenerate Perturbation Theory, 206
10.4 TimeDependent 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 TwoDimensional Harmonic Oscillator, 238
11.3.1 The Simulation of a Quantum Dot, 238
Exercises, 244
References, 244
12. Finding Eigenfunctions Using TimeDomain 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 TwoDimensional 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 OneDimensional 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