《Foundations of Quantum Physics》Burkhardt Foundations of quantum physics，Charles E. Burkhardt · Jacob J. Leventhal

Charles e. burkhardt. Jacob j. Leventhal Foundations of quantum Physics pringer

Charles E. Burkhardt · Jacob J. Leventhal Foundations of Quantum Physics 123

Helen, Charlie. Sarah. and Michelle Bette, Andy, Bradley. Dan, and Tina In loving memory of Steve Leventhal

Helen, Charlie, Sarah, and Michelle Bette, Andy, Bradley, Dan, and Tina In loving memory of Steve Leventhal

Preface This book is meant to be a text for a first course in quantum physics. It is assumed hat the student has had courses in Modern Physics and in mathematics through differential equations. The book is otherwise self-contained and does not rely on et to supple throughout except for those topics for which atomic units are especially convenient. It is our belief that for a physics major a quantum physics textbook should be more than a one- or two-semester acquaintance. Consequently, this book contains material that, while germane to the subject, the instructor might choose to omit because of time limitations. There are topics and examples included that are not normally covered in introductory textbooks. These topics are not necessarily too advanced, they are simply not usually covered. We have not, however, presumed to tell the instructor which topics must be included and which may be omitted. It is our intention that omitted subjects are available for future reference in a book that is already familiar to its owner. In short, it is our hope that the student will use the book as a reference after having completed the course. We have included at the end of most chapters a"Retrospective"of the chapter This is not meant to be merely a summary, but, rather, an overview of the importance of the material and its place in the context of previous and forthcoming chapters. For example, the Retrospective in Chapter 3 we feel is particularly important because in our experience, students spend so much time learning about eigenstates that they et the impression that physical systems "live"in eigenstates We believe that students should, after a very brief review of salient experiments and concepts that led to contemporary quantum physics( Chapter 1), begin solv ing problems. That is, the formal aspects of quantum physics, operator formalism, should be introduced only after the student has seen quantum mechanics in action This is certainly not a new approach, but we prefer it to the alternative of the for- mal mathematical introduction followed by problem solving. More importantly, we believe that the students benefit from this approach. To this end we begin with a derivation(read: rationalization) of the Schrodinger equation in Chapter 2. This chapter continues with a discussion of the nature of the solutions of the Schrodinger equation, particularly the wave function. We discuss at length both the utility of the wave function and its characteristics. It is our observation that the art of sketching wave functions has been neglected. We are led to this conclusion from discussions

Preface This book is meant to be a text for a first course in quantum physics. It is assumed that the student has had courses in Modern Physics and in mathematics through differential equations. The book is otherwise self-contained and does not rely on outside resources such as the internet to supplement the material. SI units are used throughout except for those topics for which atomic units are especially convenient. It is our belief that for a physics major a quantum physics textbook should be more than a one- or two-semester acquaintance. Consequently, this book contains material that, while germane to the subject, the instructor might choose to omit because of time limitations. There are topics and examples included that are not normally covered in introductory textbooks. These topics are not necessarily too advanced, they are simply not usually covered. We have not, however, presumed to tell the instructor which topics must be included and which may be omitted. It is our intention that omitted subjects are available for future reference in a book that is already familiar to its owner. In short, it is our hope that the student will use the book as a reference after having completed the course. We have included at the end of most chapters a “Retrospective” of the chapter. This is not meant to be merely a summary, but, rather, an overview of the importance of the material and its place in the context of previous and forthcoming chapters. For example, the Retrospective in Chapter 3 we feel is particularly important because, in our experience, students spend so much time learning about eigenstates that they get the impression that physical systems “live” in eigenstates. We believe that students should, after a very brief review of salient experiments and concepts that led to contemporary quantum physics (Chapter 1), begin solv￾ing problems. That is, the formal aspects of quantum physics, operator formalism, should be introduced only after the student has seen quantum mechanics in action. This is certainly not a new approach, but we prefer it to the alternative of the for￾mal mathematical introduction followed by problem solving. More importantly, we believe that the students benefit from this approach. To this end we begin with a derivation (read: rationalization) of the Schr¨odinger equation in Chapter 2. This chapter continues with a discussion of the nature of the solutions of the Schr¨odinger equation, particularly the wave function. We discuss at length both the utility of the wave function and its characteristics. It is our observation that the art of sketching wave functions has been neglected. We are led to this conclusion from discussions vii

with graduate students who have had the undergraduate course, but are unable to sketch wave functions for an arbitrarily drawn potential energy function. We think that such a skill is crucial for understanding quantum mechanics at the introductory level and, thus, we spend a good deal of Chapter 2 discussing qualitative aspects of the wave function In Chapter 3 we solve the Schrodinger equation for two of the most important po- tential energy functions, the infinite square well and the harmonic oscillator. A point of contrast between the these potentials is penetration of oscillator wave functions into the classically forbidden region. We discuss this penetration at length because, in our experience, students have a great deal of difficulty with this concept. We then elaborate upon this concept by presenting the details of a problem not often seen in elementary texts, an infinite square well with a barrier in the middle. This affords at, for energies less than the barrier height, the part can be found on either side of the classically impenetrable barrier, thus making the article's presence inside the barrier undeniable. This problem also sets the stage for solution of the more conventional barrier penetration problems in Chapter 5 In Chapter 4 we discuss time-dependent states. We choose to do this at this poi contrast these states with those studied in the previous chapter. While we discuss the free particle wave packet(as does virtually every other text), we also present wave packets under the influence of a constant force and of a harmonic force. This discussion will, we believe, relate nicely to a later presentation of harmonic oscilla- tor coherent states( Chapter 7) Chapter 5 is an extension of Chapter 3 in that we solve the time-independer Schrodinger equation for several different one-dimensional potential energies. In- cluded is one of the most successful analytic potential energy functions for charac- terizing diatomic molecular vibrations, the Morse potential. The chapter concludes with the WKB method for approximating solutions Chapter 6 presents the formalism of quantum physics, the mechanic tum mechanics, including a set of postulates For completeness we also discuss the Schrodinger and Heisenberg pictures. Chapter 7 is devoted to the operator solu tion of the Schrodinger equation for the harmonic oscillator with emphasis on the properties of the ladder operators. Harmonic oscillator coherent states are also dis- cussed. Chapter 8 introduces three-dimensional problems and is devoted to angular momentum. It is emphasized in this chapter that the concept of angular momen- tum in quantum mechanics transcends three-dimensional rotations (orbital angular momentum) Chapters 9 and 10 are devoted to solving the radial Schrodinger equation for everal different central potentials. In addition to the common central potentials Chapter 9 includes a thorough discussion of the isotropic harmonic oscillator using the shell model of the nucleus as an example. The isotropic oscillator also permits introduction the concept of accidental degeneracy. Because they are constituents of oscillator eigenfunctions, an attempt is made to decrypt the different conventions that are used for Laguerre polynomials and associated Laguerre polynomials In our experience, this is a source of confusion to many students. Also contained this chapter is an elaboration on the Morse potential in which three-dimensional

viii Preface with graduate students who have had the undergraduate course, but are unable to sketch wave functions for an arbitrarily drawn potential energy function. We think that such a skill is crucial for understanding quantum mechanics at the introductory level and, thus, we spend a good deal of Chapter 2 discussing qualitative aspects of the wave function. In Chapter 3 we solve the Schr¨odinger equation for two of the most important po￾tential energy functions, the infinite square well and the harmonic oscillator. A point of contrast between the these potentials is penetration of oscillator wave functions into the classically forbidden region. We discuss this penetration at length because, in our experience, students have a great deal of difficulty with this concept. We then elaborate upon this concept by presenting the details of a problem not often seen in elementary texts, an infinite square well with a barrier in the middle. This affords the opportunity to see that, for energies less than the barrier height, the particle can be found on either side of the classically impenetrable barrier, thus making the particle’s presence inside the barrier undeniable. This problem also sets the stage for solution of the more conventional barrier penetration problems in Chapter 5. In Chapter 4 we discuss time-dependent states. We choose to do this at this point to contrast these states with those studied in the previous chapter. While we discuss the free particle wave packet (as does virtually every other text), we also present wave packets under the influence of a constant force and of a harmonic force. This discussion will, we believe, relate nicely to a later presentation of harmonic oscilla￾tor coherent states (Chapter 7). Chapter 5 is an extension of Chapter 3 in that we solve the time-independent Schr¨odinger equation for several different one-dimensional potential energies. In￾cluded is one of the most successful analytic potential energy functions for charac￾terizing diatomic molecular vibrations, the Morse potential. The chapter concludes with the WKB method for approximating solutions. Chapter 6 presents the formalism of quantum physics, the mechanics of quan￾tum mechanics, including a set of postulates. For completeness we also discuss the Schr¨odinger and Heisenberg pictures. Chapter 7 is devoted to the operator solu￾tion of the Schr¨odinger equation for the harmonic oscillator with emphasis on the properties of the ladder operators. Harmonic oscillator coherent states are also dis￾cussed. Chapter 8 introduces three-dimensional problems and is devoted to angular momentum. It is emphasized in this chapter that the concept of angular momen￾tum in quantum mechanics transcends three-dimensional rotations (orbital angular momentum). Chapters 9 and 10 are devoted to solving the radial Schr¨odinger equation for several different central potentials. In addition to the common central potentials, Chapter 9 includes a thorough discussion of the isotropic harmonic oscillator using the shell model of the nucleus as an example. The isotropic oscillator also permits introduction the concept of accidental degeneracy. Because they are constituents of oscillator eigenfunctions, an attempt is made to decrypt the different conventions that are used for Laguerre polynomials and associated Laguerre polynomials. In our experience, this is a source of confusion to many students. Also contained in this chapter is an elaboration on the Morse potential in which three-dimensional

Preface Ix molecular motion is considered through rotation-vibration coupling. The discus- sion of the hydrogen atom, the sole content of Chapter 10, is standard, but, as for the isotropic oscillator, accidental degeneracy is stressed. Chapter 1l is included to demonstrate to the student that there are angular momenta in quantum mechanics other than orbital and spin angular momenta. It includes the introduction of the Lenz vector, its consequences and ramifications. This subject is not usually covered at the introductory level, but it is certainly not beyond the beginning student The material in the remaining four chapters depends heavily upon approxima tion methods. Chapter 12 presents time-independent approximation methods, while Chapter 13 illustrates the use of these methods to solve problems of physical in- terest. One problem that is included in Chapter 13, albeit superficially, is the effect of fine structure on the shell model of the nucleus. Chapter 14 treats the Stark and Zeeman effects. Particular attention is paid to the consequences of breaking the spherical symmetry of central potentials by application of an external field. Chapter 15 presents time-dependent approximation methods, followed by a discussion of tomic radiation including the einstein coefficients There are more than two hundred problems. a detailed solutions manual is avail- ble. There are a number of appendixes to the book, including the answers to all problems for which one is required. Among the other appendixes is one listing the Greek alphabet with notations on common usage of these symbols in the book There is also a short table of acronyms used in the book. The remaining appendixes contain material that is intended to be quick reference material and helpful with the core material in the book. a list of( the inevitable) corrections can be found at http://users.stlcc.edu/cburkhardt/andhttp://www.umsl.edu/-ijl/homepage/ We are indebted to several people, without whose help this manuscript would not have been completed. Helen and Charles Burkhardt, parents, read the manuscript critically. Discussions with Dr J. D. Kelley were invaluable, as was his critical read- ing of the manuscript. Professor. S. T. Manson also read the manuscript and made many useful suggestions. Discussions with Dr. M. J. Kernan were very helpful, as were her suggestions. To all of these people we offer our sincere thanks. Charles e. Burkhardt

Preface ix molecular motion is considered through rotation–vibration coupling. The discus￾sion of the hydrogen atom, the sole content of Chapter 10, is standard, but, as for the isotropic oscillator, accidental degeneracy is stressed. Chapter 11 is included to demonstrate to the student that there are angular momenta in quantum mechanics other than orbital and spin angular momenta. It includes the introduction of the Lenz vector, its consequences and ramifications. This subject is not usually covered at the introductory level, but it is certainly not beyond the beginning student. The material in the remaining four chapters depends heavily upon approxima￾tion methods. Chapter 12 presents time-independent approximation methods, while Chapter 13 illustrates the use of these methods to solve problems of physical in￾terest. One problem that is included in Chapter 13, albeit superficially, is the effect of fine structure on the shell model of the nucleus. Chapter 14 treats the Stark and Zeeman effects. Particular attention is paid to the consequences of breaking the spherical symmetry of central potentials by application of an external field. Chapter 15 presents time-dependent approximation methods, followed by a discussion of atomic radiation including the Einstein coefficients. There are more than two hundred problems. A detailed solutions manual is avail￾able. There are a number of appendixes to the book, including the answers to all problems for which one is required. Among the other appendixes is one listing the Greek alphabet with notations on common usage of these symbols in the book. There is also a short table of acronyms used in the book. The remaining appendixes contain material that is intended to be quick reference material and helpful with the core material in the book. A list of (the inevitable) corrections can be found at: http://users.stlcc.edu/cburkhardt/ and http://www.umsl.edu/∼jjl/homepage/. We are indebted to several people, without whose help this manuscript would not have been completed. Helen and Charles Burkhardt, parents, read the manuscript critically. Discussions with Dr. J. D. Kelley were invaluable, as was his critical read￾ing of the manuscript. Professor. S. T. Manson also read the manuscript and made many useful suggestions. Discussions with Dr. M. J. Kernan were very helpful, as were her suggestions. To all of these people we offer our sincere thanks. Charles E. Burkhardt Jacob J. Leventhal

Conten Preface Introduction 1. 1 Early Experiments 1.1.1 The Photoelectric Effect The Franck-Hertz Experiment 1.1.3 Atomic Spectroscopy 111357 Electron Diffraction Experiment 1.1.5 The Compton Effect 1.2 Early Theory 1.2.1 The Bohr Atom and the Correspondence Principle. 10 12.2 The de broglie wavelength 1.2.3 The Uncertainty Principle Wavelength revisited 12.5 he Classical radius of the electron 1.3 Units Retrospective References Problems 2 Elementary Wave Mechanics 2. 1 What is Doing the waving? 2.2 A Gedanken Experiment--Electron Diffraction Revisited The Wave Function 2.4 Finding the Wave Function-the Schrodinger Equationo 2.5 The Equation of Continuity 2.6 Separation of the Schrodinger Equation--Eigenfunctions 2.7 The General Solution to the Schrodinger Equation 8 Stationary States and Bound States Characteristics of the Eigenfunctions yn (x) 2.10 Retrospective Problems

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction .................................................. 1 1.1 Early Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The Franck–Hertz Experiment . . . . . . . . . . . . . . . . . 3 1.1.3 Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Electron Diffraction Experiments . . . . . . . . . . . . . . . 7 1.1.5 The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Early Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 The Bohr Atom and the Correspondence Principle . 10 1.2.2 The de Broglie Wavelength . . . . . . . . . . . . . . . . . . . . 18 1.2.3 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . 19 1.2.4 The Compton Wavelength Revisited . . . . . . . . . . . . 21 1.2.5 The Classical Radius of the Electron . . . . . . . . . . . . 23 1.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 Elementary Wave Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1 What is Doing the Waving? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 A Gedanken Experiment—Electron Diffraction Revisited . . . . . . 27 2.3 The Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Finding the Wave Function—the Schr¨odinger Equation¨o. . . . . . . 29 2.5 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Separation of the Schr¨odinger Equation—Eigenfunctions . . . . . . 33 2.7 The General Solution to the Schr¨odinger Equation . . . . . . . . . . . . 35 2.8 Stationary States and Bound States . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 Characteristics of the Eigenfunctions ψn (x) . . . . . . . . . . . . . . . . . 38 2.10 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 xi

Contents 3 Quantum Mechanics in One Dimension--Bound States I 3.1 Simple Solutions of the Schrodinger Equation The Infinite Square Well-the "Particle-in-a-Box. 47 3.1.2 The Harmonic Oscillator 56 3.2 Penetration of the Classically Forbidden Region 3.2.1 The Infinite Square Well with a Rectangular Barrier Inside 3.3 Retrospective 3.4 References Problems Time-Dependent States in One Dimension 4.1 The Ehrenfest Equation 4.2 The Free Particle Quantum Representation of Par ve 4.3.1 Momentum Representation of the Operatorx 580 4.3.2 The dirac 8-function 4.3.3 Parseval's Theorem The harmonic oscillator revisited-Momentum 4.5 Motion of a Wave Packet 4.5.1 Case l. the free Packet/Particle 4.5.2 Case Il. The Packet/Particle Subjected to a Constant Field 4.5.3 Case Ill. The Packet/Particle Subjected to a Harmonic oscillator Potential 6 Retrospective Problems 5 Stationary States in One Dimension II 3 5.1 The Potential barrier The Potential Step 5.3 The Finite Square Well-Bound States 123 5.4 The morse potential 5.5 The Linear potential 139 5.6 The WKB Approximation 5.6.1 The Nature of the Approximation 5.6 The Connection Formulas for Bound States 148 5.6.3 A Bound State Example--the Linear Potential .. 155 5.6.4 Tunneling 158 omparison with a Rectangular Barrier A Tunneling Example--Predissociation References 65 Problems

xii Contents 3 Quantum Mechanics in One Dimension—Bound States I . . . . . . . . . . . 47 3.1 Simple Solutions of the Schr¨odinger Equation . . . . . . . . . . . . . . . 47 3.1.1 The Infinite Square Well—the “Particle-in-a-Box” . 47 3.1.2 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Penetration of the Classically Forbidden Region . . . . . . . . . . . . . . 69 3.2.1 The Infinite Square Well with a Rectangular Barrier Inside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4 Time-Dependent States in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 The Ehrenfest Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Quantum Representation of Particles—Wave Packets . . . . . . . . . 86 4.3.1 Momentum Representation of the Operator x . . . . . 90 4.3.2 The Dirac δ-function . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.3 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 The Harmonic Oscillator Revisited—Momentum Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1 Case I. The Free Packet/Particle . . . . . . . . . . . . . . . . 98 4.5.2 Case II. The Packet/Particle Subjected to a Constant Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5.3 Case III. The Packet/Particle Subjected to a Harmonic Oscillator Potential . . . . . . . . . . . . . . . . . . 104 4.6 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Stationary States in One Dimension II . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 The Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 The Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 The Finite Square Well—Bound States . . . . . . . . . . . . . . . . . . . . . 123 5.4 The Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 The Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6 The WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.6.1 The Nature of the Approximation . . . . . . . . . . . . . . . 145 5.6.2 The Connection Formulas for Bound States . . . . . . 148 5.6.3 A Bound State Example—the Linear Potential . . . . 155 5.6.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.6.5 Comparison with a Rectangular Barrier . . . . . . . . . . 162 5.6.6 A Tunneling Example—Predissociation . . . . . . . . . 163 5.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 The mechanics of Quantum Mechanics 169 6.1 Abstract Vector Spaces 169 Matrix Representation of a Vector 6.1.2 Dirac notation for a vector 172 6.1.3 Operators in Quantum Mechanics The Eigenvalue Equation 179 Properties of Hermitian Operators and the Eigenvalue Equation 6.2.2 Properties of Commutators 6.3 The Postulates of Quantum Mechanics Listing of the postulates 6.3.2 Discussion of the postulates Further Consequences of the Postulates 6.4 Relation between the state Vector and the wave function.200 6.5 The Heisenberg Picture 202 6.6 Spreading of wa 6.6.1 Spreading in the Heisenberg Picture Spreading in the Schrodinger Picture 211 6.7 Retrospective 216 References 217 Problems 7 Harmonic Oscillator Solution Using Operator Methods 7.1 The Algebraic Method 219 7.1.1 The Schrodinger Picture 7.1.3 he Heisenberg Picture 7.2 Coherent States of the harmonic oscillator 7.3 Retrospective Reference 236 Problems 237 8 Quantum Mechanics in Three Dimensions-Angular Momentum .. 239 8.1 Commutation Relations 240 8.2 Angular Momentum Ladder Operators Definitions and Commutation relations 8.2.2 Angular Momentum Eigenvalues 8.3 Vector Operators 247 Orbital Angular Momentum Eigenfunctions--Spherical Harmonics 249 8.4.1 e Addition Theorem for Spherical Harmonics. 257 Parity 8.4.3 The Rigid Rotor Another Form of Angular Momentum-S

Contents xiii 6 The Mechanics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.1 Matrix Representation of a Vector . . . . . . . . . . . . . . 171 6.1.2 Dirac Notation for a Vector . . . . . . . . . . . . . . . . . . . . 172 6.1.3 Operators in Quantum Mechanics . . . . . . . . . . . . . . . 173 6.2 The Eigenvalue Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2.1 Properties of Hermitian Operators and the Eigenvalue Equation . . . . . . . . . . . . . . . . . . 180 6.2.2 Properties of Commutators . . . . . . . . . . . . . . . . . . . . 186 6.3 The Postulates of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . 189 6.3.1 Listing of the Postulates . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.2 Discussion of the Postulates . . . . . . . . . . . . . . . . . . . 190 6.3.3 Further Consequences of the Postulates . . . . . . . . . . 198 6.4 Relation Between the State Vector and the Wave Function . . . . . 200 6.5 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.6 Spreading of Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.6.1 Spreading in the Heisenberg Picture . . . . . . . . . . . . 207 6.6.2 Spreading in the Schr¨odinger Picture . . . . . . . . . . . . 211 6.7 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7 Harmonic Oscillator Solution Using Operator Methods . . . . . . . . . . . . 219 7.1 The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.1.1 The Schr¨odinger Picture . . . . . . . . . . . . . . . . . . . . . . 219 7.1.2 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.1.3 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . 227 7.2 Coherent States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . 229 7.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.4 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8 Quantum Mechanics in Three Dimensions—Angular Momentum . . . 239 8.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.2 Angular Momentum Ladder Operators . . . . . . . . . . . . . . . . . . . . . . 241 8.2.1 Definitions and Commutation Relations . . . . . . . . . 241 8.2.2 Angular Momentum Eigenvalues . . . . . . . . . . . . . . . 242 8.3 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.4 Orbital Angular Momentum Eigenfunctions—Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.4.1 The Addition Theorem for Spherical Harmonics . . 257 8.4.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.4.3 The Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8.5 Another Form of Angular Momentum—Spin . . . . . . . . . . . . . . . . 262

Contents 8.5 Matrix Representation of the Spin Operators and Ei The Stern-Gerlach Experiment 8.6 Addition of angular momenta 273 Examples of Angular Momentum Coupling Spin and Identical Particles 8.7 The Vector Model of Angular momentum Retrospective 8.9 References Problems .294 9 Central Potentials .1 Separation of the Schrodinger Equation 9.1.1 The Effective Potential 9.1.2 generacy 302 9.1.3 Behavior of the Wave Function for Small and Large 9.2 The Free Particle in Three dimensions 9.3 The Infinite Spherical Square Well 308 9. 4 The Finite Spherical Square Well 9.5 The Isotropic Harmonic Oscillator 316 Cartesian Coordinates 9.5.2 Spherical Coordinates 319 9. 6 The Morse Potential in Three Dimensions 339 9.7 Retrospective 343 References Problems 10 The hydrogen atom 347 10.1 The Radial Equation--Energy Eigenvalues 10.2 Degeneracy of the Energy Eigenvalues 10.3 The Radial Equation--Energy Eigenfunctions 10.4 The Complete Energy Eigenfunctions 10.5 Retrospective 362 10.6 References Problems 11 Angular momentum-Encore 365 11.1 The Classical Kepler Problem 11.2 The Quantum Mechanical Kepler Problem 11.3 The Action of A+ 11.4 Retrospective 11.5 References 372 Problems 372

xiv Contents 8.5.1 Matrix Representation of the Spin Operators and Eigenkets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 8.5.2 The Stern–Gerlach Experiment . . . . . . . . . . . . . . . . . 270 8.6 Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.6.1 Examples of Angular Momentum Coupling . . . . . . 277 8.6.2 Spin and Identical Particles . . . . . . . . . . . . . . . . . . . . 285 8.7 The Vector Model of Angular Momentum . . . . . . . . . . . . . . . . . . 292 8.8 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 9 Central Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.1 Separation of the Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . 298 9.1.1 The Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . 300 9.1.2 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 9.1.3 Behavior of the Wave Function for Small and Large Values of r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.2 The Free Particle in Three Dimensions . . . . . . . . . . . . . . . . . . . . . 305 9.3 The Infinite Spherical Square Well . . . . . . . . . . . . . . . . . . . . . . . . . 308 9.4 The Finite Spherical Square Well . . . . . . . . . . . . . . . . . . . . . . . . . . 309 9.5 The Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.5.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 317 9.5.2 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 319 9.6 The Morse Potential in Three Dimensions . . . . . . . . . . . . . . . . . . . 339 9.7 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 10 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 10.1 The Radial Equation—Energy Eigenvalues . . . . . . . . . . . . . . . . . . 347 10.2 Degeneracy of the Energy Eigenvalues. . . . . . . . . . . . . . . . . . . . . . 352 10.3 The Radial Equation—Energy Eigenfunctions . . . . . . . . . . . . . . . 354 10.4 The Complete Energy Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 361 10.5 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 10.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 11 Angular Momentum—Encore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 11.1 The Classical Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 11.2 The Quantum Mechanical Kepler Problem . . . . . . . . . . . . . . . . . . 367 11.3 The Action of Aˆ + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 11.4 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 11.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

12 Time-Independent Approximation Methods 375 12.1 Perturbation Theory 375 Nondegenerate Perturbation Theory 12.1.2 Degenerate Perturbation Theory 12.2 The Variational Method Problems 13 Applications of Time-Independent Approximation Methods 13.1 Hydrogen Atoms Breaking the Degeneracy--Fine Structure 13.2 Spin-Orbit Coupling and the Shell Model of the Nucleus 13. 3 Helium Atoms 411 The Ground State 411 3.3.2 Excited States 13. 4 Multielectron Atoms 422 13.5 Retrospective 13.6 References Problems 14 Atoms in external fields 14.1 Hydrogen Atoms in External Fields 431 4.1.1 Electric fields the Stark effect 431 2 Magnetic Fields-The Zeeman Effect 14.2 Multielectron Atoms in External Magnetic Fields 442 14.3 Retrospective 46 14.4 References 446 Problems 15 Time-Dependent perturbations 449 15.1 Time Dependence of the State Vector 15.2 Two-State Systems 452 Harmonic Perturbation--Rotating Wave pproximation 452 15.2.2 Constant Perturbation Turned On att=o 455 15.3 Time-Dependent Perturbation Theory 457 15.4 Two-state Systems Using Perturbation Theory Harmonic perturbation 15.4.2 Constant Perturbation Turned On att=0 15.5 Extension to Multistate Systems 464 15.5.1 Harmonic perturbation 15.5.2 Constant perturbation Turned On att=o 465 15.5.3 ransitions to a Continuum of states-The Golden rule 15.6 Interactions of Atoms with Radiation 468 15.6.1 The Nature of Electromagnetic transitions

Contents xv 12 Time-Independent Approximation Methods . . . . . . . . . . . . . . . . . . . . . . 375 12.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 12.1.1 Nondegenerate Perturbation Theory . . . . . . . . . . . . . 375 12.1.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . 382 12.2 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 13 Applications of Time-Independent Approximation Methods . . . . . . . . 397 13.1 Hydrogen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 13.1.1 Breaking the Degeneracy—Fine Structure . . . . . . . . 397 13.2 Spin–Orbit Coupling and the Shell Model of the Nucleus . . . . . . 409 13.3 Helium Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.3.1 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.3.2 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 13.4 Multielectron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 13.5 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 13.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 14 Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 14.1 Hydrogen Atoms in External Fields . . . . . . . . . . . . . . . . . . . . . . . . 431 14.1.1 Electric Fields—the Stark Effect . . . . . . . . . . . . . . . . 431 14.1.2 Magnetic Fields—The Zeeman Effect . . . . . . . . . . . 436 14.2 Multielectron Atoms in External Magnetic Fields . . . . . . . . . . . . 442 14.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 14.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 15 Time-Dependent Perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 15.1 Time Dependence of the State Vector . . . . . . . . . . . . . . . . . . . . . . . 449 15.2 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 15.2.1 Harmonic Perturbation—Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 15.2.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 455 15.3 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . 457 15.4 Two-state Systems Using Perturbation Theory . . . . . . . . . . . . . . . 459 15.4.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 459 15.4.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 462 15.5 Extension to Multistate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 15.5.1 Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . . . . . 464 15.5.2 Constant Perturbation Turned On at t = 0 . . . . . . . . 465 15.5.3 Transitions to a Continuum of States—The Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 15.6 Interactions of Atoms with Radiation . . . . . . . . . . . . . . . . . . . . . . . 468 15.6.1 The Nature of Electromagnetic Transitions . . . . . . . 469