《Foundations of Quantum Physics》Burkhardt Foundations of quantum physics，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