This page intentionally left blank www.com Fundamentals of Quantum Mechanics Quantum mechanics has evolved from a subject of study in pure physics to one with a wide range of applications in many diverse fields. The basic concepts of quantum mechanics are explained in this book in a concise and easy-to-read manner, leading toward applications in solid state electronics and modern optics. Following a logical sequence, the book is focused on the key ideas and is conceptually and mathematically self-contained. The fundamental principles of quantum mechanics are illustrated by showing their application to systems such as the hydrogen atom, multi-electron ions and atoms, the formation of simple organic molecules and crystalline solids of prac- tical importance.
It leads on from these basic concepts to discuss some of the most important applications in modern semiconductor electronics and optics. Containing many homework problems, the book is suitable for senior-level under- graduate and graduate level students in electrical engineering, materials science, and applied physics and chemistry. Tang is the Spencer T. Olin Professor of Engineering at Cornell University, Ithaca, NY.
His research interest has been in quantum electronics, nonlinear optics, femtosecond optics and ultrafast process in molecules and semiconductors, and he has published extensively in these fields. He is a Fellow of the IEEE, the Optical Society of America, and the Americal Physical Society, and is a member of the US National Academy of Engineering. He was the winner of the Charles H. Townes Award of the Optical Society of America in 1996.com Fundamentals of Quantum Mechanics For Solid State Electronics and Optics C.
TANG Cornell University, Ithaca, NY www.com cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www.org/9780521829526 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 isbn-13 978-0-511-12595-9 eBook (NetLibrary) isbn-10 0-511-12595-x eBook (NetLibrary) isbn-13 978-0-521-82952-6 hardback isbn-10 0-521-82952-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com To Louise www.com Contents Preface page x 1 Classical mechanics vs.1 Brief overview of classical mechanics 1 1.2 Overview of quantum mechanics 2 2 Basic postulates and mathematical tools 8 2.3 Equations of motion (Postulate 3) 18 2.4 Eigen functions, basis states, and representations 21 2.5 Alternative notations and formulations 23 2.6 Problems 31 3 Wave/particle duality and de Broglie waves 33 3.1 Free particles and de Broglie waves 33 3.2 Momentum representation and wave packets 37 3.3 Problems 39 4 Particles at boundaries, potential steps, barriers, and in quantum wells 40 4.1 Boundary conditions and probability currents 40 4.2 Particles at a potential step, up or down 43 4.3 Particles at a barrier and the quantum mechanical tunneling effect 47 4.4 Quantum wells and bound states 50 4.5 Three-dimensional potential box or quantum well 59 4.6 Problems 60 5 The harmonic oscillator and photons 63 5.1 The harmonic oscillator based on Heisenberg’s formulation of quantum mechanics 63 5.2 The harmonic oscillator based on Schrödinger’s formalism 70 5.3 Superposition state and wave packet oscillation 73 5.5 Problems 84 vii www.com viii Contents 6 The hydrogen atom 86 6.1 The Hamiltonian of the hydrogen atom 86 6.2 Angular momentum of the hydrogen atom 87 6.3 Solution of the time-independent Schrödinger equation for the hydrogen atom 94 6.4 Structure of the hydrogen atom 97 6.5 Electron spin and the theory of generalized angular momentum 101 6.6 Spin–orbit interaction in the hydrogen atom 106 6.7 Problems 108 7 Multi-electron ions and the periodic table 110 7.1 Hamiltonian of the multi-electron ions and atoms 110 7.2 Solutions of the time-independent Schrödinger equation for multi- electron ions and atoms 112 7.3 The periodic table 115 7.4 Problems 118 8 Interaction of atoms with electromagnetic radiation 119 8.1 Schrödinger’s equation for electric dipole interaction of atoms with electromagnetic radiation 119 8.2 Time-dependent perturbation theory 120 8.4 Selection rules and the spectra of hydrogen atoms and hydrogen-like ions 126 8.5 The emission and absorption processes 128 8.6 Light Amplification by Stimulated Emission of Radiation (LASER) and the Einstein A- and B-coefficients 130 8.7 Problems 133 9 Simple molecular orbitals and crystalline structures 135 9.1 Time-independent perturbation theory 135 9.2 Covalent bonding of diatomic molecules 139 9.3 sp, sp2, and sp3 orbitals and examples of simple organic molecules 144 9.4 Diamond and zincblende structures and space lattices 148 9.5 Problems 149 10 Electronic properties of semiconductors and the p-n junction 151 10.1 Molecular orbital picture of the valence and conduction bands of semiconductors 151 10.2 Nearly-free-electron model of solids and the Bloch theorem 153 10.3 The k-space and the E vs.4 Density-of-states and the Fermi energy for the free-electron gas model 163 10.5 Fermi–Dirac distribution function and the chemical potential 164 10.6 Effective mass of electrons and holes and group velocity in semiconductors 170 www.com Contents ix 10.7 n-type and p-type extrinsic semiconductors 173 10.9 Problems 180 11 The density matrix and the quantum mechanic Boltzmann equation 182 11.1 Definitions of the density operator and the density matrix 182 11.2 Physical interpretation and properties of the density matrix 183 11.3 The density matrix equation or the quantum mechanic Boltzmann equation 186 11.4 Examples of the solutions and applications of the density matrix equations 188 11.5 Problems 202 References 204 Index 205 www.com Preface Quantum mechanics has evolved from a subject of study in pure physics to one with a vast range of applications in many diverse fields. Some of its most important applica- tions are in modern solid state electronics and optics.
As such, it is now a part of the required undergraduate curriculum of more and more electrical engineering, materials science, and applied physics schools. This book is based on the lecture notes that I have developed over the years teaching introductory quantum mechanics to students at the senior/first year graduate school level whose interest is primarily in applications in solid state electronics and modern optics. There are many excellent introductory text books on quantum mechanics for students majoring in physics or chemistry that emphasize atomic and nuclear physics for the former and molecular and chemical physics for the latter. Often, the approach is to begin from a historic perspective, recounting some of the experimental observa- tions that could not be explained on the basis of the principles of classical mechanics and electrodynamics, followed by descriptions of various early attempts at developing a set of new principles that could explain these ‘anomalies.’ It is a good way to show the students the historical thinking that led to the discovery and formulation of the basic principles of quantum mechanics.
This might have been a reasonable approach in the first half of the twentieth century when it was an interesting story to be told and people still needed to be convinced of its validity and utility. Most students today know that quantum theory is now well established and important. What they want to know is not how to reinvent quantum mechanics, but what the basic principles are concisely and how they are used in applications in atomic, molecular, and solid state physics. For electronics, materials science, and applied physics students in particular, they need to see, above all, how quantum mechanics forms the foundations of modern semiconductor electronics and optics.
To meet this need is then the primary goal of this introductory text/reference book, for such students and for those who did not have any quantum mechanics in their earlier days as an undergraduate student but wish now to learn the subject on their own. This book is not encyclopedic in nature but is focused on the key concepts and results. Hopefully it makes sense pedagogically. As a textbook, it is conceptually and mathematically self-contained in the sense that all the results are derived, or derivable, from first principles, based on the material presented in the book in a logical order without excessive reliance on reference sources.
The emphasis is on concise physical x www.com Preface xi explanations, complemented by rigorous mathematical demonstrations, of how things work and why they work the way they do. A brief introduction is given in Chapter 1 on how one goes about formulating and solving problems on the atomic and subatomic scale. This is followed in Chapter 2 by a concise description of the basic postulates of quantum mechanics and the terminology and mathematical tools that one will need for the rest of the book. This part of the book by necessity tends to be on the abstract side and might appear to be a little formal to some of the beginning students.
It is not necessary to master all the mathematical details and complications at this stage. For organizational reasons, I feel that it is better to collect all this information at one place at the beginning so that the flow of thoughts and the discussions of the main subject matter will not be repeatedly interrupted later on by the need to introduce the language and tools needed. The basic principles of quantum mechanics are then applied to a number of simple prototype problems in Chapters 3–5 that help to clarify the basic concepts and as a preparation for discussing the more realistic physical problems of interest in later chapters.4 on photons is a discussion of the application of the basic theory of harmonic oscillators to radiation oscillators. It gives the basic rules of quantization of electromagnetic fields and discusses the historically important problem of black- body radiation and the more recently developed quantum theory of coherent optical states.
For an introductory course on quantum mechanics, this material can perhaps be skipped. Chapters 6 and 7 deal with the hydrogenic and multi-electron atoms and ions. Since the emphasis of this book is not on atomic spectroscopy, some of the mathematical details that can be found in many of the excellent books on atomic physics are not repeated in this book, except for the key concepts and results. These chapters form the foundations of the subsequent discussions in Chapter 8 on the important topics of time-dependent perturbation theory and the interaction of radiation with matter.
It naturally leads to Einstein’s theory of resonant absorption and emission of radiation by atoms. One of its most important progeny is the ubiquitous optical marvel known as the LASER (Light Amplification by Stimulated Emission of Radiation). From the hydrogenic and multi-electron atoms, we move on to the increasingly more complicated world of molecules and solids in Chapter 9. The increased complex- ity of the physical systems requires more sophisticated approximation procedures to deal with the related mathematical problems.
The basic concept and methodology of time-independent perturbation theory is introduced and applied to covalent-bonded diatomic and simple organic molecules. Crystalline solids are in some sense giant molecules with periodic lattice structures. Of particular interest are the sp3-bonded elemental and compound semiconductors of diamond and zincblende structures. Some of the most important applications of quantum mechanics are in semi- conductor physics and technology based on the properties of charge-carriers in periodic lattices of ions.
Basic concepts and results on the electronic properties of semiconductors are discussed in Chapter 10. The molecular-orbital picture and the nearly-free-electron model of the origin of the conduction and valence bands in semiconductors based on the powerful Bloch theorem are developed. From these www.com xii Preface follow the commonly used concepts and parameters to describe the dynamics of charge-carriers in semiconductors, culminating finally in one of the most important building blocks of modern electronic and optical devices: the p–n junction.