Quantum Mechanicsfor Scientists and EngineersCambridgeDavid A. B. Miller
To Pat, Andrew, and Susan
Quantum Mechanics for Scientistsand EngineersDavid A. B. MillerStanford Universitywww.cambridge.org© Cambridge University PressCambridge University Press978-0-521-89783-9 - Quantum Mechanics for Scientists and EngineersDavid A. B. MillerCopyright InformationMore information
cambridge university pressCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ ao Paulo, DelhiCambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USAwww.cambridge.orgInformation on this title: www.cambridge.org/9780521897839C ?Cambridge University Press 2008This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published 2008Printed in the United States of AmericaA catalog record for this publication is available from the British Library.Library of Congress Cataloging in Publication DataMiller, D. A. B.Quantum mechanics for scientists and engineers / David A. B. Miller.p. cm.Includes bibliographical references and index.ISBN 978-0-521-89783-9 (hardback)1. Quantum theory.I. Title.QC174.13.M556 2008530.12–dc222008001249ISBN978-0-521-89783-9 hardbackCambridge University Press has no responsibility forthe persistence or accuracy of URLs for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is, or will remain, accurate or appropriate.www.cambridge.org© Cambridge University PressCambridge University Press978-0-521-89783-9 - Quantum Mechanics for Scientists and EngineersDavid A. B. MillerCopyright InformationMore information
Contents Preface xiii How to use this book xvi Chapter 1 1.1 1.2 1.3 Introduction Quantum mechanics and real life Quantum mechanics as an intellectual achievement Using quantum mechanics 1 1 4 6 Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 Waves and quantum mechanics – Schrödinger’s equation 8 Rationalization of Schrödinger’s equation Probability densities Diffraction by two slits Linearity of quantum mechanics: multiplying by a constant Normalization of the wavefunction Particle in an infinitely deep potential well (“particle in a box”) Properties of sets of eigenfunctions Particles and barriers of finite heights Particle in a finite potential well Harmonic oscillator Particle in a linearly varying potential Summary of concepts 8 11 12 16 17 18 23 26 32 39 42 50 Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 The time-dependent Schrödinger equation Rationalization of the time-dependent Schrödinger equation Relation to the time-independent Schrödinger equation Solutions of the time-dependent Schrödinger equation Linearity of quantum mechanics: linear superposition Time dependence and expansion in the energy eigenstates Time evolution of infinite potential well and harmonic oscillator Time evolution of wavepackets Quantum mechanical measurement and expectation values The Hamiltonian Operators and expectation values Time evolution and the Hamiltonian operator Momentum and position operators Uncertainty principle Particle current Quantum mechanics and Schrödinger’s equation Summary of concepts 54 55 57 58 59 60 61 67 73 77 78 79 81 83 86 88 89
viii Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 Functions and operators Functions as vectors Vector space Operators Linear operators Evaluating the elements of the matrix associated with an operator Bilinear expansion of linear operators Specific important types of linear operators Identity operator Inverse operator Unitary operators Hermitian operators Matrix form of derivative operators Matrix corresponding to multiplying by a function Summary of concepts 94 95 101 104 105 108 109 111 111 114 114 120 125 126 126 Chapter 5 5.1 5.2 5.3 5.4 5.5 Operators and quantum mechanics Commutation of operators General form of the uncertainty principle Transitioning from sums to integrals Continuous eigenvalues and delta functions Summary of concepts 131 131 133 137 138 152 Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Approximation methods in quantum mechanics Example problem – potential well with an electric field Use of finite matrices Time-independent non-degenerate perturbation theory Degenerate perturbation theory Tight binding model Variational method Summary of concepts 156 157 159 163 172 174 178 182 Chapter 7 7.1 7.2 7.3 7.4 7.5 Time-dependent perturbation theory Time-dependent perturbations Simple oscillating perturbations Refractive index Nonlinear optical coefficients Summary of Concepts 184 184 187 194 197 207 Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Quantum mechanics in crystalline materials Crystals One electron approximation Bloch theorem Density of states in k-space Band structure Effective mass theory Density of states in energy Densities of states in quantum wells 209 209 211 211 215 216 218 222 223
ix 8.9 8.10 8.11 k.p method Use of Fermi’s Golden Rule Summary of Concepts 228 233 241 Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 Angular momentum Angular momentum operators L squared operator Visualization of spherical harmonic functions Comments on notation Visualization of angular momentum Summary of concepts 244 244 249 252 255 256 257 Chapter 10 The hydrogen atom 10.1 Multiple particle wavefunctions 10.2 Hamiltonian for the hydrogen atom problem 10.3 Coordinates for the hydrogen atom problem 10.4 Solving for the internal states of the hydrogen atom 10.5 Solutions of the hydrogen atom problem 10.6 Summary of concepts 259 260 261 263 266 272 277 Chapter 11 Methods for one-dimensional problems 11.1 Tunneling probabilities 11.2 Transfer matrix 11.3 Penetration factor for slowly varying barriers 11.4 Electron emission with a potential barrier 11.5 Summary of Concepts 279 279 282 290 292 297 Chapter 12 Spin 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 299 300 302 304 305 307 309 309 310 Angular momentum and magnetic moments State vectors for spin angular momentum Operators for spin angular momentum The Bloch sphere Direct product spaces and wavefunctions with spin Pauli equation Where does spin come from? Summary of concepts Chapter 13 Identical particles 13.1 Scattering of identical particles 13.2 Pauli exclusion principle 13.3 States, single-particle states, and modes 13.4 Exchange energy 13.5 Extension to more than two identical particles 13.6 Multiple particle basis functions 13.7 Thermal distribution functions 13.8 Important extreme examples of states of multiple identical particles 13.9 Quantum mechanical particles reconsidered 13.10 Distinguishable and indistinguishable particles 313 313 317 318 318 323 325 330 331 332 333
x 13.11 Summary of concepts 334 Chapter 14 The density matrix 14.1 Pure and mixed states 14.2 Density operator 14.3 Density matrix and ensemble average values 14.4 Time-evolution of the density matrix 14.5 Interaction of light with a two-level “atomic” system 14.6 Density matrix and perturbation theory 14.7 Summary of concepts 337 337 340 341 343 345 352 353 Chapter 15 Harmonic oscillators and photons 15.1 Harmonic oscillator and raising and lowering operators 15.2 Hamilton’s equations and generalized position and momentum 15.3 Quantization of electromagnetic fields 15.4 Nature of the quantum mechanical states of an electromagnetic mode 15.5 Field operators 15.6 Quantum mechanical states of an electromagnetic field mode 15.7 Generalization to sets of modes 15.8 Vibrational modes 15.9 Summary of concepts 356 356 361 363 368 369 372 375 380 381 Chapter 16 Fermion operators 16.1 Postulation of fermion annihilation and creation operators 16.2 Wavefunction operator 16.3 Fermion Hamiltonians 16.4 Summary of concepts 385 386 395 397 406 Chapter 17 Interaction of different kinds of particles 17.1 States and commutation relations for different kinds of particles 17.2 Operators for systems with different kinds of particles 17.3 Perturbation theory with annihilation and creation operators 17.4 Stimulated emission, spontaneous emission, and optical absorption 17.5 Summary of concepts 408 408 409 411 413 424 Chapter 18 Quantum information 18.1 Quantum mechanical measurements and wavefunction collapse 18.2 Quantum cryptography 18.3 Entanglement 18.4 Quantum computing 18.5 Quantum teleportation 18.6 Summary of concepts 426 426 427 433 436 439 442 Chapter 19 Interpretation of quantum mechanics 19.1 Hidden variables and Bell’s inequalities 19.2 The measurement problem 19.3 Solutions to the measurement problem 19.4 Epilogue 443 443 450 451 456
xi 19.5 Summary of concepts 457 Appendix A Background mathematics A.1 Geometrical vectors A.2 Exponential and logarithm notation A.3 Trigonometric notation A.4 Complex numbers A.5 Differential calculus A.6 Differential equations A.7 Summation notation A.8 Integral calculus A.9 Matrices A.10 Product notation A.11 Factorial 459 459 462 463 463 466 470 476 477 481 492 492 Appendix B Background physics B.1 Elementary classical mechanics B.2 Electrostatics B.3 Frequency units B.4 Waves and diffraction 493 493 496 497 497 Appendix C Vector calculus C.1 Vector calculus operators C.2 Spherical polar coordinates C.3 Cylindrical coordinates C.4 Vector calculus identities 501 501 506 508 509 Appendix D Maxwell’s equations and electromagnetism D.1 Polarization of a material D.2 Maxwell’s equations D.3 Maxwell’s equations in free space D.4 Electromagnetic wave equation in free space D.5 Electromagnetic plane waves D.6 Polarization of a wave D.7 Energy density D.8 Energy flow D.9 Modes 511 511 512 514 514 515 516 516 516 518 Appendix E Perturbing Hamiltonian for optical absorption E.1 Justification of the classical Hamiltonian E.2 Quantum mechanical Hamiltonian E.3 Choice of gauge E.4 Approximation to linear system 521 521 522 523 524
xii Appendix F Early history of quantum mechanics 525 Appendix G Some useful mathematical formulae G.1 Elementary mathematical expressions G.2 Formulae for sines, cosines, and exponentials G.3 Special functions 527 527 528 531 Appendix H Greek alphabet 535 Appendix I Fundamental constants 536 Bibliography 537 Memorization list 541 Index 545
Preface This book introduces quantum mechanics to scientists and engineers. It can be used as a text for junior undergraduates onwards through to graduate students and professionals. The level and approach are aimed at anyone with a reasonable scientific or technical background looking for a solid but accessible introduction to the subject. The coverage and depth are substantial enough for a first quantum mechanics course for physicists. At the same time, the level of required background in physics and mathematics has been kept to a minimum to suit those also from other science and engineering backgrounds. Quantum mechanics has long been essential for all physicists and in other physical science subjects such as chemistry. With the growing interest in nanotechnology, quantum mechanics has recently become increasingly important for an ever-widening range of engineering disciplines, such as electrical and mechanical engineering, and for subjects such as materials science that underlie many modern devices. Many physics students also find that they are increasingly motivated in the subject as the everyday applications become clear. Non-physicists have a particular problem in finding a suitable introduction to the subject. The typical physics quantum mechanics course or text deals with many topics that, though fundamentally interesting, are useful primarily to physicists doing physics; that choice of topics also means omitting many others that are just as truly quantum mechanics, but have more practical applications. Too often, the result is that engineers or applied scientists cannot afford the time or cannot sustain the motivation to follow such a physics-oriented sequence. As a result, they never have a proper grounding in the subject. Instead, they pick up bits and pieces in other courses or texts. Learning quantum mechanics in such a piecemeal approach is especially difficult; the student then never properly confronts the many fundamentally counterintuitive concepts of the subject. Those concepts need to be understood quite deeply if the student is ever going to apply the subject with any reliability in any novel situation. Too often also, even after working hard in a quantum mechanics class, and even after passing the exams, the student is still left with the depressing feeling that they do not understand the subject at all. To address the needs of its broad intended readership, this book differs from most others in three ways. First, it presumes as little as possible in prior knowledge of physics. Specifically, it does not presume the advanced classical mechanics (including concepts such as Hamiltonians and Lagrangians) that is often a prerequisite in physics quantum mechanics texts and courses. Second, in two background appendices, it summarizes all of the key physics and mathematics beyond the high-school level that the reader needs to start the subject. Third, it introduces the quantum mechanics that underlies many important areas of application, including semiconductor physics, optics, and optoelectronics. Such areas are usually omitted from quantum mechanics texts, but this book introduces many of the quantum mechanical principles and models that are exploited in those subjects. It is also my belief and experience that using quantum mechanics in several different and practical areas of application removes many of the difficulties in understanding the subject. If quantum mechanics is only illustrated through examples that are found in the more esoteric
xiv branches of physics, the subject itself can seem irrelevant and obscure. There is nothing like designing a real device with quantum mechanics to make the subject tangible and meaningful. Even with its deliberately limited prerequisites and its increased discussion of applications, this book offers a solid foundation in the subject. That foundation should prepare the reader well for the quantum mechanics in either advanced physics or in deeper study of practical applications in other scientific and engineering fields. The emphasis in the book is on understanding the ideas and techniques of quantum mechanics rather than attempting to cover all possible examples of their use. A key goal of this book is that the reader should subsequently be able to pick up texts in a broad range of areas, including, for example, advanced quantum mechanics for physicists, solid state and semiconductor physics and devices, optoelectronics, quantum information, and quantum optics, and find they already have all the necessary basic tools and conceptual background in quantum mechanics to make rapid progress. It is possible to teach quantum mechanics in many different ways, though most sequences will start with Schrödinger’s wave equation and work forward from there. Even though the final emphasis in this book may be different from some other quantum mechanics courses, I have deliberately chosen not to take a radical approach here. This is for three reasons: first, most college and university teachers will be most comfortable with a relatively standard approach since that is the one they have most probably experienced themselves; second, taking a core approach that is relatively conventional will make it easier for readers (and teachers) to connect with the many other good physics quantum mechanics books; third, this book should also be accessible and useful to professionals who have previously studied quantum mechanics to some degree, but need to update their knowledge or connect to the modern applications in engineering or applied sciences. The background requirements for the reader are relatively modest, and should represent little problem for students or professionals in engineering, applied sciences, physics, or other physical sciences. This material has been taught with apparent success to students in applied physics, electrical engineering, mechanical engineering, materials science, and other science and engineering disciplines, from 3rd year undergraduate level up to graduate students. In mathematics, the reader should have a basic knowledge in calculus, complex numbers, elementary matrix algebra, geometrical vectors, and simple and partial differential equations. In physics, the reader should be familiar with ordinary Newtonian classical mechanics and elementary electricity and magnetism. The key requirements are summarized in two background appendices in case the reader wants to refresh some background knowledge or fill in gaps. A few other pieces of physics and mathematics are introduced as needed in the main body of the text. It is helpful if the student has had some prior exposure to elementary modern physics, such as the ideas of electrons, photons, and the Bohr model of the atom, but no particular results are presumed here. The necessary parts of Hamiltonian classical mechanics will be introduced briefly when required in later Chapters. This book goes deeper into certain subjects, such as the quantum mechanics of light, than most introductory physics texts. For the later Chapters on the quantum mechanics of light, additional knowledge of vector calculus and electromagnetism to the level of Maxwell’s equations are presumed, though again these are summarized in appendices. One intent of the book is for the student to acquire a strong understanding of the concepts of quantum mechanics at the level beyond mere mathematical description. As a result, I have chosen to try to explain concepts with limited use of mathematics wherever possible. With the ready availability of computers and appropriate software for numerical calculations and
xv simulations, it is progressively easier to teach principles of quantum mechanics without as heavy an emphasis on analytical techniques. Such numerical approaches are also closer to the methods that an engineer will likely use for calculations in real problems anyway, and access to some form of computer and high-level software package is assumed for some of the problems. This approach substantially increases the range of problems that can be examined both for tutorial examples and for applications. Finally, I will make one personal statement on handling the conceptual difficulties of quantum mechanics in texts and courses. Some texts are guilty of stating quantum mechanical postulates, concepts and assumptions as if they should be obvious, or at least obviously acceptable, when in fact they are far from obvious even to experienced practitioners or teachers. In many cases, these are subjects of continuing debate at the highest level. I try throughout to be honest about those concepts and assumptions that are genuinely unclear as to their obviousness or even correctness. I believe it is a particularly heinous sin to pretend that some concept should be clear to the student when it is, in fact, not even clear to the professor (an overused technique that preserves professorial ego at the expense of the student’s!). It is a pleasure to acknowledge the many teaching assistants who have provided much useful feedback and correction of my errors in this material as I have taught it at Stanford, including Aparna Bhatnagar, Julien Boudet, Eleni Diamanti, Onur Fidaner, Martina Gerken, Noah Helman, Ekin Kocabas, Bianca Nelson, Tomas Sarmiento, and Scott Sharpe. I would like to thank Ingrid Tarien for much help in preparing many parts of the course material, and Marjorie Ford for many helpful comments on writing. I am also pleased to acknowledge my many professorial colleagues at Stanford, including Steve Harris, Walt Harrison, Jelena Vuckovic, and Yoshi Yamamoto in particular, for many stimulating, informative, and provocative discussions about quantum mechanics. I would especially like to thank Jelena Vuckovic, who successfully taught the subject to many students despite having to use much of this material as a course reader, and who consequently corrected numerous errors and clarified many points. All remaining errors and shortcomings are, of course, my sole responsibility, and any further corrections and suggestions are most welcome. David A. B. Miller Stanford, California, September 2007
How to use this book For teachers The entire material in this book could be taught in a one-year course. More likely, depending on the interests and goals of the teacher and students, and the length of time available, only some of the more advanced topics will be covered in detail. In a two-quarter course sequence for senior undergraduates and for engineering graduate students at Stanford, the majority of the material here will be covered, with a few topics omitted and some covered in lesser depth. The core material (Chapters 1 – 5) on Schrödinger’s equation and on the mathematics behind quantum mechanics should be taught in any course. Chapter 4 gives a more explicit introduction to the ideas of linear operators than is found in most texts. Chapter 4 also explains and introduces Dirac notation, which is used from that point onwards in the book. This introduction of Dirac notation is earlier than in many older texts, but it saves considerable time thereafter in describing quantum mechanics. Experience teaching engineering students in particular, most of whom are quite familiar with linear algebra and matrices from other applications in engineering, shows that they have no difficulties with this concept. Aside from that core there are many possible choices about the sequence of material and on what material needs to be included in a course. The prerequisites for each Chapter are clearly stated at the beginning of the Chapter. There are also some Sections in several of the Chapters that are optional, or that may only need to be read through when first encountered. These Sections are clearly marked. The discussion of methods for one-dimensional problems in Chapter 11 can come at any point after the material on Schrödinger’s equations (Chapters 2 and 3). The core transfer matrix part could even be taught directly after the time-independent equation (Chapter 2). The material is optional in that it is not central to later topics, but in my experience students usually find it stimulating and empowering to be able to do calculations with simple computer programs based on these methods. This can make the student comfortable with the subject, and begin to give them some intuitive feel for many quantum mechanical phenomena. (These methods are also used in practice for the design of real optoelectronic devices.) For a broad range of applications, the approximation methods of quantum mechanics (Chapter 6 and 7) are probably the next most important after Chapters 1 - 5. The specific topic of the quantum mechanics of crystalline materials (Chapter 8) is a particularly important topic for many applications, and can be introduced at any point after Chapter 7; it is not, however, required for subsequent Chapters (except for a few examples, and some optional parts at the end of Chapter 11), so the teacher can choose how far he or she wants to progress through this Chapter. For fundamentals, angular momentum (Chapter 9) and the hydrogen atom (Chapter 10) are the next most central topics, both of which can be taught directly after Chapter 5 if desired. After these, the next most important fundamental topics are spin (Chapter 12) and identical particles (Chapter 13), and these should probably be included in the second quarter or semester if not before.
xvii Chapter 14 introduces the important technique of the density matrix for connecting to statistical mechanics, and it can be introduced at any point after Chapter 5; preferably the student would also have covered Chapters 6 and 7 so they are familiar with perturbation theory, though that is not required. The density matrix material is not required for subsequent Chapters, so this Chapter is optional. The sequence of Chapters 15 – 17 introduces the quantum mechanics of electromagnetic fields and light, and also the important technique of second quantization in general, including fermion operators (a technique that is also used extensively in more advanced solid state physics). The inclusion of this material on the quantum mechanics of light is the largest departure from typical introductory quantum mechanics texts. It does however redress a balance in material that is important from a practical point of view; we cannot describe even the simplest light emitter (including an ordinary light bulb.) or light detector without it, for example. This material is also very substantial quantum mechanics at the next level of the subject. These Chapters do require almost all of the preceding material, with the possible exceptions of Chapters 8, 11, and 14. The final two Chapters, Chapter 18 on a brief introduction to quantum information concepts and Chapter 19 on the interpretation of quantum mechanics, could conceivably be presented with only Chapters 1 – 5 as prerequisites. Preferably also Chapters 9, 10, 12, and 13 would have been covered, and it is probably a good idea that the student has been working with quantum mechanics successfully for some time before attempting to grapple with the tricky conceptual and philosophical aspects in these final Chapters. The material in these Chapters is well suited to the end of a course, when it is often unreasonable to include any further new material in a final exam, but yet one wants to keep the students’ interest with stimulating ideas. Problems are introduced directly after the earliest possible Sections rather than being deferred to the ends of the Chapters, thus giving the greatest flexibility in assigning homework. Some problems can be used as substantial assignments, and all such problems are clearly marked. These can be used as “take-home” problems or exams, or as extended exercises coupled with tutorial “question and answer” sessions. These assignments may necessarily involve some more work, such as significant amounts of (relatively straightforward) algebra or calculations with a computer. I have found, though, that students gain a much greater confidence in the subject once they have used it for something beyond elementary exercises, exercises that are necessarily often artificial. At least, these assignments tend to approach the subject from the point of view of a problem to be solved rather than an exercise that just uses the last technique that was studied. Some of these larger assignments deal with quite realistic uses of quantum mechanics. At the very end of the book, I also include a suggested list of simple formulae to be memorized in each Chapter. These lists could also be used as the basis of simple quizzes, or as required learning for “closed-book” exams. For students Necessary background Students will come to this book with very different backgrounds. You may recently have studied a lot of physics and mathematics at college level. If so, then you are ready to start. I suggest you have a quick look at Appendices A and B just to see the notations used in this book before starting Chapter 2.
xviii For others, your mathematical or physics background may be less complete, or it may be some time since you have seen or used some of the relevant parts of these subjects. Rest assured, first of all, that in writing this book I have presumed the least possible knowledge of mathematics and physics consistent with teaching quantum mechanics, and much less than the typical quantum mechanics text requires. Ideally, I expect you have had the physics and mathematics typical of first or second year college level for general engineering or physical science students. You do absolutely have to know elementary algebra, calculus, and physics to a good pre-college level, however. I suggest you read the Background Mathematics Appendix A and the Background Physics Appendix B to see if you understand most of that. If not too much of that is new to you, then you should be able to proceed into the main body of this book. If you find some new topics in these Appendices, there is in principle enough material there to “patch over” those holes in knowledge temporarily so that you can use the mathematics and physics needed to start quantum mechanics; these Appendices are not, however, meant to be a substitute for learning these topics in greater depth.. Study aids in this book Lists of concepts introduced Because there are many concepts that the student needs to understand in quantum mechanics, I have summarized the most important ones at the end of the Chapters in which they are introduced. These summaries should help both in following the “plot” of the book, and in revising the material. Appendices The book is as reasonably self-contained as I can make it. In addition to the background Appendices A and B covering the overall prerequisite mathematics and physics, additional background material needed later on is introduced in Appendices C and D (vector calculus and electromagnetism), and one specific detailed derivation is given in Appendix E. Appendix F summarizes the early history of quantum mechanics, Appendix G collects and summarizes most of the mathematical formulae that will be needed in the book, including the most useful ones from elementary algebra, trigonometric functions, and calculus. Appendix H gives the Greek alphabet (every single letter of it is used somewhere in quantum mechanics), and Appendix I lists all the relevant fundamental constants. Problems There are about 160 problems and assignments, collected at the ends of the earliest possible Sections rather than at the ends of the Chapters. Memorization list Quantum mechanics, like many aspects of physics, is not primarily about learning large numbers of formulae, but rather understanding the key concepts clearly and deeply. It will, however, save a lot of time (including in exams!) to learn a few basic formulae by heart, and certainly if you also understand these well, you should have a good command of the subject. At the very end of the book, there is a list of formulae worth memorizing in each Chapter of the book. None of these formulae is particularly complicated – the most complicated ones are the Schrödinger wave equation in its two forms. Many of the formulae are simply short definitions of key mathematical concepts. If you learn these formulae chapter by chapter as you work through the book, there are not very many formulae to learn at any one time.
xix The list here is not of the formulae themselves, but rather is of descriptions of them so you can use this list as an exercise to test how successfully you have learned these key results. Self-teaching If you are teaching yourself quantum mechanics using this book, first of all, congratulations to you for having the courage to tackle what most people typically regard as a daunting subject. For someone with elementary college level physics and mathematics, I believe it is quite an accessible subject in fact. But, the most important point is that you must not start learning quantum mechanics “on the fly” by picking and choosing just the bits you need from this book or any other. Trying to learn quantum mechanics like that would be like trying to learn a language by reading a dictionary. You cannot treat quantum mechanics as just a set of formulae to be substituted into problems, just as you cannot translate a sentence from one language to another just by looking up the individual words in a dictionary and writing down their translations. There are just so many counterintuitive aspects about quantum mechanics that you will never understand it in that piecemeal way, and most likely you would not use the formulae correctly anyway. Make yourself work on all of the first several Chapters, through at least Chapter 5; that will get you to a first plateau of understanding. You can be somewhat more selective after that. For the next level of understanding, you need to study angular momentum, spin and identical particles (Chapters 9, 12, and 13). Which other Chapters you use will depend on your interests or needs. Of course, it is worthwhile studying all of them if you have the time! Especially if you have no tutor of whom you can ask questions, then I also expect that you should be looking at other quantum mechanics books as well. Use this one as your core, and when I have just not managed to explain something clearly enough...