Quantum Mechanics for Electrical Engineers

by Sullivan, Dennis M.

• ISBN13: 9780470874097
• ISBN10: 0470874090
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-01-24
• Publisher: Wiley-IEEE Press

Summary

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.

Author Biography

Dennis M. Sullivan is Professor of Electrical and Computer Engineering at the University o1 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

Table of Contents

Preface	p. xiii
Acknowledgments	p. xv
About the Author	p. xvii
Introduction	p. 1
Why Quantum Mechanics?	p. 1
Photoelectric Effect	p. 1
Wave-Particle Duality	p. 2
Energy Equations	p. 3
The Schrödinger Equation	p. 5
Simulation of the One-Dimensional, Time-Dependent Schrödinger Equation	p. 7
Propagation of a Particle in Free Space	p. 8
Propagation of a Particle Interacting with a Potential	p. 11
Physical Parameters: The Observables	p. 14
The Potential V(x)	p. 17
The Conduction Band of a Semiconductor	p. 17
A Particle in an Electric Field	p. 17
Propagating through Potential Barriers	p. 20
Summary	p. 23
Exercises	p. 24
References	p. 25
Stationary States	p. 27
The Infinite Well	p. 28
Eigenstates and Eigenenergies	p. 30
Quantization	p. 33
Eigenfunction Decomposition	p. 34
Periodic Boundary Conditions	p. 38
Eigenfunctions for Arbitrarily Shaped Potentials	p. 39
Coupled Wells	p. 41
Bra-ket Notation	p. 44
Summary	p. 47
Exercises	p. 47
References	p. 49
Fourier Theory in Quantum Mechanics	p. 51
The Fourier Transform	p. 51
Fourier Analysis and Available States	p. 55
Uncertainty	p. 59
Transmission via FFT	p. 62
Summary	p. 66
Exercises	p. 67
References	p. 69
Matrix Algebra in Quantum Mechanics	p. 71
Vector and Matrix Representation	p. 71
State Variables as Vectors	p. 71
Operators as Matrices	p. 73
Matrix Representation of the Hamiltonian	p. 76
Finding the Eigenvalues and Eigenvectors of a Matrix	p. 77
A Well with Periodic Boundary Conditions	p. 77
The Harmonic Oscillator	p. 80
The Eigenspace Representation	p. 81
Formalism	p. 83
Hermitian Operators	p. 83
Function Spaces	p. 84
Appendix: Review of Matrix Algebra	p. 85
Exercises	p. 88
References	p. 90
A Brief Introduction to Statistical Mechanics	p. 91
Density of States	p. 91
One-Dimensional Density of States	p. 92
Two-Dimensional Density of States	p. 94
Three-Dimensional Density of States	p. 96
The Density of States in the Conduction Band of a Semiconductor	p. 97
Probability Distributions	p. 98
Fermions versus Classical Particles	p. 98
Probability Distributions as a Function of Energy	p. 99
Distribution of Fermion Balls	p. 101
Particles in the One-Dimensional Infinite Well	p. 105
Boltzmann Approximation	p. 106
The Equilibrium Distribution of Electrons and Holes	p. 107
The Electron Density and the Density Matrix	p. 110
The Density Matrix	p. 111
Exercises	p. 113
References	p. 114
Bands and Subbands	p. 115
Bands in Semiconductors	p. 115
The Effective Mass	p. 118
Modes (Subbands) in Quantum Structures	p. 123
Exercises	p. 128
References	p. 129
The Schrödinger Equation for Spin-1/2 Fermions	p. 131
Spin in Fermions	p. 131
Spinors in Three Dimensions	p. 132
The Pauli Spin Matrices	p. 135
Simulation of Spin	p. 136
An Electron in a Magnetic Field	p. 142
A Charged Particle Moving in Combined E and B Fields	p. 146
The Hartree-Fock Approximation	p. 148
The Hartree Term	p. 148
The Fock Term	p. 153
Exercises	p. 155
References	p. 157
The Green's Function Formulation	p. 159
Introduction	p. 160
The Density Matrix and the Spectral Matrix	p. 161
The Matrix Version of the Green's Function	p. 164
Eigenfunction Representation of Green's Function	p. 165
Real Space Representation of Green's Function	p. 167
The Self-Energy Matrix	p. 169
An Electric Field across the Channel	p. 174
A Short Discussion on Contacts	p. 175
Exercises	p. 176
References	p. 176
Transmission	p. 177
The Single-Energy Channel	p. 177
Current Flow	p. 179
The Transmission Matrix	p. 181
Flow into the Channel	p. 183
Flow out of the Channel	p. 184
Transmission	p. 185
Determining Current Flow	p. 186
Conductance	p. 189
Büttiker Probes	p. 191
A Simulation Example	p. 194
Exercises	p. 196
References	p. 197
Approximation Methods	p. 199
The Variational Method	p. 199
Nondegenerate Perturbation Theory	p. 202
First-Order Corrections	p. 203
Second-Order Corrections	p. 206
Degenerate Perturbation Theory	p. 206
Time-Dependent Perturbation Theory	p. 209
An Electric Field Added to an Infinite Well	p. 212
Sinusoidal Perturbations	p. 213
Absorption, Emission, and Stimulated Emission	p. 215
Calculation of Sinusoidal Perturbations Using Fourier Theory	p. 216
Fermi's Golden Rule	p. 221
Exercises	p. 223
References	p. 225
The Harmonic Oscillator	p. 227
The Harmonic Oscillator in One Dimension	p. 227
Illustration of the Harmonic Oscillator Eigenfunctions	p. 232
Compatible Observables	p. 233
The Coherent State of the Harmonic Oscillator	p. 233
The Superposition of Two Eigentates in an Infinite Well	p. 234
The Superposition of Four Eigenstates in a Harmonic Oscillator	p. 235
The Coherent State	p. 236
The Two-Dimensional Harmonic Oscillator	p. 238
The Simulation of a Quantum Dot	p. 238
Exercises	p. 244
References	p. 244
Finding Eigenfunctions Using Time-Domain Simulation	p. 245
Finding the Eigenenergies and Eigenfunctions in One Dimension	p. 245
Finding the Eigenfunctions	p. 248
Finding the Eigenfunctions of Two-Dimensional Structures	p. 249
Finding the Eigenfunctions in an Irregular Structure	p. 252
Finding a Complete Set of Eigenfunctions	p. 257
Exercises	p. 259
References	p. 259
Important Constants and Units	p. 261
Fourier Analysis and the Fast Fourier Transform (FFT)	p. 265
The Structure of the FFT	p. 265
Windowing	p. 267
FFT of the State Variable	p. 270
Exercises	p. 271
References	p. 271
An Introduction to the Green's Function Method	p. 273
A One-Dimensional Electromagnetic Cavity	p. 275
Exercises	p. 279
References	p. 279
Listings of the Programs Used in this Book	p. 281
Chapter 1	p. 281
Chapter 2	p. 284
Chapter 3	p. 295
Chapter 4	p. 309
Chapter 5	p. 312
Chapter 6	p. 314
Chapter 7	p. 323
Chapter 8	p. 336
Chapter 9	p. 345
Chapter 10	p. 356
Chapter 11	p. 378
Chapter 12	p. 395
Appendix B	p. 415
Index	p. 419
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