Quantum Mechanics: Foundations and Applications

Transcription

1 Arno Böhm Quantum Mechanics: Foundations and Applications Third Edition, Revised and Enlarged Prepared with Mark Loewe With 96 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

2 Contents Preface to the Third Edition Preface to the Second Edition Acknowledgments vii ix xiii CHAPTER I Mathematical Preliminaries The Mathematical Language of Quantum Mechanics Linear Spaces, Scalar Product Linear Operators Basis Systems and Eigenvector Decomposition Realizations of Operators and of Linear Spaces Hermite Polynomials as an Example of Orthonormal Basis Functions 28 Appendix to Section Continuous Functionals How the Mathematical Quantities Will Be Used 39 Problems 39 CHAPTER II Foundations of Quantum Mechanics The Harmonic Oscillator 43 II. 1 Introduction The First Postulate of Quantum Mechanics Algebra of the Harmonic Oscillator The Relation Between Experimental Data and Quantum-Mechanical Observables The Basic Assumptions Applied to the Harmonic Oscillator, and Some Historical Remarks Some General Consequences of the Basic Assumptions of Quantum Mechanics Eigenvectors of Position and Momentum Operators; the Wave Functions of the Harmonic Oscillator 84 xv

3 xvi Contents 11.8 Postulates II and III for Observables with Continuous Spectra Position and Momentum Measurements Particles and Waves 101 Problems 112 CHAPTER III Energy Spectra of Some Molecules 117 ULI Transitions Between Energy Levels of Vibrating Molecules The Limitations of the Oscillator Model The Rigid Rotator The Algebra of Angular Momentum Rotation Spectra Combination of Quantum Physical Systems The Vibrating Rotator 146 Problems 155 CHAPTER IV Complete Systems of Commuting Observables 159 CHAPTER V Addition of Angular Momenta The Wigner-Eckart Theorem 164 V.l Introduction The Elementary Rotator 164 V.2 Combination of Elementary Rotators 165 V.3 Tensor Operators and the Wigner-Eckart Theorem 176 Appendix to Section V V.4 Parity 192 Problem 204 CHAPTER VI Hydrogen Atom The Quantum-Mechanical Kepler Problem 205 VI. 1 Introduction 205 VI.2 Classical Kepler Problem 206 VI.3 Quantum-Mechanical Kepler Problem 208 VI.4 Properties of the Algebra of Angular Momentum and the Lenz Vector 213 VI.5 The Hydrogen Spectrum 215 Problem 222 CHAPTER VII Alkali Atoms and the Schrödinger Equation of One-Electron Atoms 223 VII. 1 The Alkali Hamiltonian and Perturbation Theory 223 VII.2 Calculation of the Matrix Elements of the Operator Q~ v 227 VII.3 Wave Functions and Schrödinger Equation of the Hydrogen Atom and the Alkali Atoms 234 Problem 241 CHAPTER VIII Perturbation Theory 242 VIII.1 Perturbation of the Discrete Spectrum 242 VIII.2 Perturbation of the Continuous Spectrum The Lippman-Schwinger Equation 248 Problems 251

4 Contents xvii CHAPTER IX Electron Spin 253 IX. 1 Introduction 253 IX.2 The Fine Structure Qualitative Considerations 255 IX.3 Fine-Structure Interaction 261 IX.4 Fine Structure of Atomic Spectra 268 IX.5 Selection Rules 270 IX.6 Remarks on the State of an Electron in Atoms 271 Problems 272 CHAPTER X Indistinguishable Particles 274 X.l Introduction 274 Problem 281 CHAPTER XI Two-Electron Systems The Helium Atom 282 XI. 1 The Two Antisymmetric Subspaces of the Helium Atom 282 XI.2 Discrete Energy Levels of Helium 287 XI.3 Selection Rules and Singlet-Triplet Mixing for the Helium Atom 297 XI.4 Doubly Excited States of Helium 303 Problems 309 CHAPTER XII Time Evolution 310 XII.1 Time Evolution 31 XII.A Mathematical Appendix: Definitions and Properties of Operators that Depend upon a Parameter 324 Problems 326 CHAPTER XIII Some Fundamental Properties of Quantum Mechanics 328 XIII. 1 Change of the State by the Dynamical Law and by the Measuring Process The Stern-Gerlach Experiment 328 Appendix to Section XIII ХШ.2 Spin Correlations in a Singlet State 342 XIII.3 Bell's Inequalities, Hidden Variables, and the Einstein-Podolsky- Rosen Paradox 347 Problems 354 CHAPTER XIV Transitions in Quantum Physical Systems Cross Section 356 XIV. 1 Introduction 356 XIV.2 Transition Probabilities and Transition Rates 358 XIV.3 Cross Sections 362 XIV.4 The Relation of Cross Sections to the Fundamental Physical Observables 365 XIV.5 Derivation of Cross-Section Formulas for the Scattering of a Beam off a Fixed Target 368 Problems 384

5 xviii Contents CHAPTER XV Formal Scattering Theory and Other Theoretical Considerations 387 XV. 1 The Lippman-Schwinger Equation 387 XV.2 In-States and Out-States 391 XV. 3 The S-Operator and the Meiler Wave Operators 399 XV.A Appendix 407 CHAPTER XVI Elastic and Inelastic Scattering for Spherically Symmetric Interactions XVI. 1 Partial-Wave Expansion XVI.2 Unitarity and Phase Shifts XVI. 3 Argand Diagrams Problems CHAPTER XVII Free and Exact Radial Wave Functions 425 XVII. 1 Introduction 425 XVII.2 The Radial Wave Equation 426 XVII.3 The Free Radial Wave Function 430 XVII.4 The Exact Radial Wave Function 432 XVII.5 Poles and Bound States 439 XVII.6 Survey of Some General Properties of Scattering Amplitudes and Phase Shifts 441 XVII.A Mathematical Appendix on Analytic Functions 444 Problems 450 CHAPTER XVIII Resonance Phenomena 452 XVIII. 1 Introduction 452 XVIII.2 Time Delay and Phase Shifts 457 XVIII.3 Causality Conditions 464 XVIII.4 Causality and Analyticity 467 XVIII.5 Brief Description of the Analyticity Properties of the S-Matrix 471 XVIII.6 Resonance Scattering Breit-Wigner Formula for Elastic Scattering 476 XVIII.7 The Physical Effect of a Virtual State 487 XVIII.8 Argand Diagrams for Elastic Resonances and Phase-Shift Analysis 489 XVIII.9 Comparison with the Observed Cross Section: The Effect of Background and Finite Energy Resolution 493 Problems 503 CHAPTER XIX Time Reversal 505 XIX. 1 Space-Inversion Invariance and the Properties of the S-Matrix 505 XIX.2 Time Reversal 507 Appendix to Section XIX XIX.3 Time-Reversal Invariance and the Properties of the S-Matrix 512 Problems 516

6 Contents XIX CHAPTER XX Resonances in Multichannel Systems 517 XX. 1 Introduction 517 XX.2 Single and Double Resonances 518 XX.3 Argand Diagrams for Inelastic Resonances 532 CHAPTER XXI The Decay of Unstable Physical Systems 537 XXI. 1 Introduction 537 XXI.2 Lifetime and Decay Rate 539 XXI.3 The Description of a Decaying State and the Exponential Decay Law 542 XXI.4 Gamow Vectors and Their Association to the Resonance Poles of the S-Matrix 549 XXI.5 The Golden Rule 563 XXI.6 Partial Decay Rates 567 Problems 569 CHAPTER XXII Quantal Phase Factors and Their Consequences 571 XXII. 1 Introduction 571 XXII.2 A Quantum Physical System in a Slowly Changing Environment 573 XXII.3 A Spinning Quantum System in a Slowly Changing External Magnetic Field The Adiabatic Approximation 587 XXII.4 A Spinning Quantum System in a Precessing External Magnetic Field The General Cyclic Evolution 598 Problems 614 CHAPTER XXIII A Quantum Physical System in a Quantum Environment The Gauge Theory of Molecular Physics 617 XXIII. 1 Introduction 617 XXIII.2 The Hamiltonian of the Diatomic Molecule 618 XXIII.3 The Born-Oppenheimer Method 623 XXIII.4 Gauge Theories 631 XXIII.5 The Gauge Theory of Molecular Physics 636 XXIII.6 The Electronic States of Diatomic Molecules 643 XXIII.7 The Monopole of the Diatomic Molecule 645 Problems 658 Epilogue 661 Bibliography 664 Index 669