Engineering Quantum Mechanics

308 Pages·2011·3.076 MB·English

by  Doyeol Ahn, Seoung-Hwan Park

ENGINEERING
QUANTUM MECHANICS
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ENGINEERING
QUANTUM MECHANICS
Doyeol Ahn
Seoung-Hwan Park
IEEE PRESS
A JOHN WILEY & SONS, INC., PUBLICATION
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Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Ahn, Doyeol.
Engineering Quantum Mechanics/Doyeol Ahn, Seoung-Hwan Park.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-10763-8
1. Quantum theory. 2. Stochastic processes. 3. Engineering mathematics.
4. Semiconductors–Electric properties–Mathematical models. I. Park, Seoung-Hwan.
II. Title.
QC174.12.A393 2011
620.001'53012–dc22
2010044304
oBook ISBN: 978-1-118-01782-1
ePDF ISBN: 978-1-118-01780-7
ePub ISBN: 978-1-118-01781-4
Printed in Singapore.
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CONTENTS
Preface vii
PART I Fundamentals 1
1 Basic Quantum Mechanics 3
1.1 Measurements and Probability 3
1.2 Dirac Formulation 4
1.3 Brief Detour to Classical Mechanics 8
1.4 A Road to Quantum Mechanics 14
1.5 The Uncertainty Principle 21
1.6 The Harmonic Oscillator 22
1.7 Angular Momentum Eigenstates 29
1.8 Quantization of Electromagnetic Fields 35
1.9 Perturbation Theory 38
Problems 41
References 43
2 Basic Quantum Statistical Mechanics 45
2.1 Elementary Statistical Mechanics 45
2.2 Second Quantization 51
2.3 Density Operators 54
2.4 The Coherent State 58
2.5 The Squeezed State 62
2.6 Coherent Interactions Between Atoms and Fields 68
2.7 The Jaynes–Cummings Model 69
Problems 71
References 72
3 Elementary Theory of Electronic Band Structure
in Semiconductors 73
3.1 Bloch Theorem and Effective Mass Theory 73
3.2 The Luttinger–Kohn Hamiltonian 84
3.3 The Zinc Blende Hamiltonian 105
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vi CONTENTS
3.4 The Wurtzite Hamiltonian 114
3.5 Band Structure of Zinc Blende and
Wurtzite Semiconductors 123
3.6 Crystal Orientation Effects on a Zinc
Blende Hamiltonian 135
3.7 Crystal Orientation Effects on a Wurtzite Hamiltonian 152
Problems 168
References 169
PART II Modern Applications 171
4 Quantum Information Science 173
4.1 Quantum Bits and Tensor Products 173
4.2 Quantum Entanglement 175
4.3 Quantum Teleportation 178
4.4 Evolution of the Quantum State:
Quantum Information Processing 180
4.5 A Measure of Information 183
4.6 Quantum Black Holes 184
Appendix A: Derivation of Equation (4.82) 202
Appendix B: Derivation of Equations (4.93) and (4.106) 203
Problems 204
References 205
5 Modern Semiconductor Laser Theory 207
5.1 Density Operator Description of Optical Interactions 209
5.2 The Time-Convolutionless Equation 211
5.3 The Theory of Non-Markovian Optical Gain in
Semiconductor Lasers 223
5.4 Optical Gain of a Quantum Well Laser with
Non-Markovian Relaxation and Many-Body Effects 232
5.5 Numerical Methods for Valence Band Structure
in Nanostructures 235
5.6 Zinc Blende Bulk and Quantum Well Structures 252
5.7 Wurtzite Bulk and Quantum Well Structures 258
5.8 Quantum Wires and Quantum Dots 265
Appendix: Fortran 77 Code for the Band Structure 274
Problems 286
References 287
Index 289
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Preface
Quantum mechanics is becoming more important in applied science
and engineering, especially with the recent developments in quantum
computing, as well as the rapid progress in optoelectronic devices. This
textbook is intended for graduate students and advanced undergradu-
ate students in electrical engineering, physics, and materials science and
engineering. It also provides the necessary theoretical background for
researchers in optoelectronics or semiconductor devices. In the task
of providing advanced instruction for both students and researchers,
quantum mechanics presents special diffi culties because of its hierar-
chical structures. The more abstract formalisms and techniques are
quite meaningless until one has mastered the earlier stages in classical
physics, which most engineering students are lacking.
Quantum mechanics has become an essential tool for modern engi-
neering. This book covers topics such as semiconductors and laser
physics, which are traditionally quantum mechanical, as well as rela-
tively new topics in the fi eld, such as quantum computation and quantum
information. These fi elds have seen an explosive growth during the past
10 years, as quantum computing or quantum information processing
can have a signifi cant impact on today ’ s electronics and computations.
The essence of quantum computing is the direct usage of the superposi-
tion and entanglement of quantum mechanics. The most challenging
research topics include the generation and manipulation of quantum
entangled systems, developing the fundamental theory of entangle-
ment, decoherence control, and the demonstration of the scalability of
quantum information processing.
In laser physics, there has been a growing interest in the model of
semiconductor lasers with non - Markovian relaxation partially because
of the dissatisfaction with the conventional model for optical gain
in predicting the correct gain spectrum and the thermodynamic rela-
tions. This is mainly due to the poor convergence properties of the
lineshape function, that is, the Lorentzian lineshape, used in the con-
ventional model. In this book, a non - Markovian model for the optical
gain of semiconductors is developed, taking into account the rigorous
electronic band structure, many - body effects, and the non - Markovian
vii
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viii PREFACE
relaxation using the quantum statistical reduced - density operator for-
malism for an arbitrary driven system coupled to a stochastic reservoir.
Example programs based on Fortran 77 will also be provided for band
structures of zinc blende quantum wells.
Many - body effects are taken into account within the time - dependent
Hartree – Fock. Various semiconductor lasers including strained - layer
quantum well lasers and wurtzite GaN blue - green quantum well lasers
are discussed.
We thank Professor Shun - Lien Chuang, Doyeol Ahn’s Ph.D. thesis
adviser, for extensive enlightening and encouragement over many
years. We are also grateful to many colleagues and friends, especially
Frank Stern, B. D. Choe, Han Jo Lim, H. S. Min, M. S. Kim, Robert
Mann, Tim Ralph, K. S. Seo, Y. S. Cho, and Chancellor Sam Bum Lee.
The support of our research by the Korean Ministry of Education,
Science and Technology is greatly appreciated. This book would not
have been completed without the patience and continued encourage-
ment of our editors at Wiley and above all the encouragement and
understanding of Taeyeon Yim and Young - Mee An. Thanks for putting
up with us.
Doyeol Ahn
Seoung-Hwan Park
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PART I
Fundamentals
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1
Basic Quantum Mechanics
1.1 MEASUREMENTS AND PROBABILITY
In the beginning of 20th century, it was discovered that the behavior of
very small particles, such as electrons, the nuclei of atoms, and mole-
cules, cannot be described by classical mechanics, which had been quite
successful in explaining the macroscopic world until then. Nonetheless,
it was soon discovered that the description of these phenomena on the
atomic scale is possible by the set of laws described by quantum mechan-
ics. Both classical mechanics and quantum mechanics are based on the
description of measurements of observable quantities called dynamical
variables, such as position, momentum, and energy. Consider an experi-
ment in which we can make three measurements successive in time.
Let ’ s denote the fi rst of observable quantities A , the second B , and the
third C . We also denote a ,b, and c as one of a number of possible results
that could come from the measurement of A , B , and C , respectively.
Let P (b|a) be the conditional probability that if the measurement of
A results in a , then the measurement of B will result in b . From the
elementary probability theory, the conditional probability P(b|a) can
be written as follows:
P(a,b)
P(b|a)= , (1.1)
P(a)
where P(a,b) is the joint probability that measurements of both A and
B will give a and b , simultaneously, and P(a) is the probability that the
measurement of A will give the outcome a. For three successive mea-
surements A , B , and C , the conditional probability P(cb|a) that if the
measurement of A results in a , then the measurement of B will result
in b , then the measurement of C will result in c is given by:
P(cb|a)=P(c|b)P(b|a). (1.2)
Engineering Quantum Mechanics, First Edition. Doyeol Ahn, Seoung-Hwan Park.
© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.
3
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4 BASIC QUANTUM MECHANICS
Moreover, if we sum Equation (1.2) over all the mutually exclusive
alternatives for b , we obtain the conditional probability P (c|a):
∑
P(c|a)= P(c|b)P(b|a).
(1.3)
b
In classical mechanics, the above relation described by Equation
(1.3) is always true. However, it was found that the above relation
sometimes fails on the atomic scale, and one needs to modify
Equations (1.1) to (1.3) by introducing new complex quantities ϕ ,
ba
ϕ , and ϕ , called probability amplitudes, which are related to proba-
cb ca
bilities by [1,2]
P(b|a)= ϕ 2, (1.4)
ba
and
2
∑
P(c|a)= ϕ ϕ . (1.5)
cb ba
b
Equations (1.4) and (1.5) describe the probability of measurement
outcome in quantum mechanics. From the mathematical point of view,
the probability amplitude is found to be the inner product of vectors
in a special kind of vector space called the Hilbert space:
ϕ = b a , (1.6)
ba
where a is the column vector, called the “ ket vector, ” corresponding
to the observable a , and b is the row vector, called the “ bra vector, ”
corresponding to the observable b .
1.2 DIRAC FORMULATION
In quantum mechanics, a physical state corresponding to the observ-
able quantity a is represented by the ket vector, a , in a complex vector
space H with dimension N . For example, when N = 2, the ket vector a
is a column vector given by [1,3]
⎛a ⎞
a =⎜ 1⎟, (1.7)
⎝ ⎠
a
2
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