Quantum mechanics for electrical engineers

434 Pages·2012·12.18 MB·english

by  Dennis Michael Sullivan

QUANTUM MECHANICS
FOR ELECTRICAL
ENGINEERS
IEEE Press
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Piscataway, NJ 08854
IEEE Press Editorial Board
Lajos Hanzo, Editor in Chief
R. Abhari M. El - Hawary O. P. Malik
J. Anderson B - M. Haemmerli S. Nahavandi
G. W. Arnold M. Lanzerotti T. Samad
F. Canavero D. Jacobson G. Zobrist
Kenneth Moore, Director of IEEE Book and Information Services (BIS)
Technical Reviewers
Prof. Richard Ziolkowski, University of Arizona
Prof. F. Marty Ytreberg, University of Idaho
Prof. David Citrin, G eorgia Institute of Technology
Prof. Steven Hughes, Q ueens University
QUANTUM MECHANICS
FOR ELECTRICAL
ENGINEERS
DENNIS M. SULLIVAN
IEEE Series on Microelectronics Systems
Jake Baker, Series Editor
IEEE PRESS
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2012 by the Institute of Electrical and Electronics Engineers, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data
ISBN: 978-0-470-87409-7
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To
My Girl

CONTENTS
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
vii
viii CONTENTS
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
CONTENTS ix
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