Quantum Mechanics: Foundations and Applications

I Mathematical Preliminaries.- I.1 The Mathematical Language of Quantum Mechanics.- I.2 Linear Spaces, Scalar Product.- I.3 Linear Operators.- I.4 Basis Systems and Eigenvector Decomposition.- I.5 Realizations of Operators and of Linear Spaces.- I.6 Hermite Polynomials as an Example of Orthonormal Basis Functions.- Appendix to Section 1.6.- I.7 Continuous Functionals.- I.8 How the Mathematical Quantities Will Be Used.- Problems.- II Foundations of Quantum Mechanics—The Harmonic Oscillator.- II.1 Introduction.- II.2 The First Postulate of Quantum Mechanics.- II.3 Algebra of the Harmonic Oscillator.- II.4 The Relation Between Experimental Data and Quantum-Mechanical Observables.- II.5 The Basic Assumptions Applied to the Harmonic Oscillator, and Some Historical Remarks.- II.6 Some General Consequences of the Basic Assumptions of Quantum Mechanics.- II.7 Eigenvectors of Position and Momentum Operators; the Wave Functions of the Harmonic Oscillator.- II.8 Postulates II and III for Observables with Continuous Spectra.- II.9 Position and Momentum Measurements—Particles and Waves.- Problems.- III Energy Spectra of Some Molecules.- III.1 Transitions Between Energy Levels of Vibrating Molecules—The Limitations of the Oscillator Model.- III.2 The Rigid Rotator.- III.3 The Algebra of Angular Momentum.- III.4 Rotation Spectra.- III.5 Combination of Quantum Physical Systems—The Vibrating Rotator.- Problems.- IV Complete Systems of Commuting Observables.- V Addition of Angular Momenta—The Wigner-Eckart Theorem.- V.1 Introduction—The Elementary Rotator.- V.2 Combination of Elementary Rotators.- V.3 Tensor Operators and the Wigner-Eckart Theorem.- Appendix to Section V.3.- V.4 Parity.- Problem.- VI Hydrogen Atom—The Quantum-Mechanical Kepler Problem.- VI.1 Introduction.- VI.2 Classical Kepler Problem.- VI.3 Quantum-Mechanical Kepler Problem.- VI.4 Properties of the Algebra of Angular Momentum and the Lenz Vector.- VI.5 The Hydrogen Spectrum.- Problem.- VII Alkali Atoms and the Schrödinger Equation of One-Electron Atoms.- VII.1 The Alkali Hamiltonian and Perturbation Theory.- VII.2 Calculation of the Matrix Elements of the Operator Q-?.- VII.3 Wave Functions and Schrödinger Equation of the Hydrogen Atom and the Alkali Atoms.- Problem.- VIII Perturbation Theory.- VIII.1 Perturbation of the Discrete Spectrum.- VIII.2 Perturbation of the Continuous Spectrum—The Lippman-Schwinger Equation.- Problems.- IX Electron Spin.- IX.1 Introduction.- IX.2 The Fine Structure—Qualitative Considerations.- IX.3 Fine-Structure Interaction.- IX.4 Fine Structure of Atomic Spectra.- IX.5 Selection Rules.- IX.6 Remarks on the State of an Electron in Atoms.- Problems.- X Indistinguishable Particles.- X.1 Introduction.- Problem.- XI Two-Electron Systems—The Helium Atom.- XI.1 The Two Antisymmetric Subspaces of the Helium Atom.- XI.2 Discrete Energy Levels of Helium.- XI.3 Selection Rules and Singlet-Triplet Mixing for the Helium Atom.- XI.4 Doubly Excited States of Helium.- Problems.- XII Time Evolution.- XII.1 Time Evolution.- XII.A Mathematical Appendix: Definitions and Properties of Operators that Depend upon a Parameter.- Problems.- XIII Some Fundamental Properties of Quantum Mechanics.- XIII.1 Change of the State by the Dynamical Law and by the Measuring Process—The Stern-Gerlach Experiment.- Appendix to Section XIII.1.- XIII.2 Spin Correlations in a Singlet State.- XIII.3 Bell’s Inequalities, Hidden Variables, and the Einstein-Podolsky-Rosen Paradox.- Problems.- XIV Transitions in Quantum Physical Systems—Cross Section.- XIV.1 Introduction.- XIV.2 Transition Probabilities and Transition Rates.- XIV.3 Cross Sections.- XIV.4 The Relation of Cross Sections to the Fundamental Physical Observables.- XIV.5 Derivation of Cross-Section Formulas for the Scattering of a Beam off a Fixed Target.- Problems.- XV Formal Scattering Theory and Other Theoretical Considerations.- XV.1 The Lippman-Schwinger Equation.- XV.2 In-States and Out-States.- XV.3 The S-Operator and the Møller Wave Operators.- XV.A Appendix.- 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.- XVII Free and Exact Radial Wave Functions.- XVII.1 Introduction.- XVII.2 The Radial Wave Equation.- XVII.3 The Free Radial Wave Function.- XVII.4 The Exact Radial Wave Function.- XVII.5 Poles and Bound States.- XVII.6 Survey of Some General Properties of Scattering Amplitudes and Phase Shifts.- XVII.A Mathematical Appendix on Analytic Functions.- Problems.- XVIII Resonance Phenomena.- XVIII.1 Introduction.- XVIII.2 Time Delay and Phase Shifts.- XVIII.3 Causality Conditions.- XVIII.4 Causality and Analyticity.- XVIII.5 Brief Description of the Analyticity Properties of the S-Matrix.- XVIII.6 Resonance Scattering—Breit-Wigner Formula for Elastic Scattering.- XVIII.7 The Physical Effect of a Virtual State.- XVIII.8 Argand Diagrams for Elastic Resonances and Phase-Shift Analysis.- XVIII.9 Comparison with the Observed Cross Section: The Effect of Background and Finite Energy Resolution.- Problems.- XIX Time Reversal.- XIX.1 Space-Inversion Invariance and the Properties of the S-Matrix.- XIX.2 Time Reversal.- Appendix to Section XIX.2.- XIX.3 Time-Reversal Invariance and the Properties of the S-Matrix.- Problems.- XX Resonances in Multichannel Systems.- XX.1 Introduction.- XX.2 Single and Double Resonances.- XX.3 Argand Diagrams for Inelastic Resonances.- XXI The Decay of Unstable Physical Systems.- XXI.1 Introduction.- XXI.2 Lifetime and Decay Rate.- XXI.3 The Description of a Decaying State and the Exponential Decay Law.- XXI.4 Gamow Vectors and Their Association to the Resonance Poles of the S-Matrix.- XXI.5 The Golden Rule.- XXI.6 Partial Decay Rates.- Problems.- XXII Quantal Phase Factors and Their Consequences.- XXII.1 Introduction.- XXII.2 A Quantum Physical System in a Slowly Changing Environment.- XXII.3 A Spinning Quantum System in a Slowly Changing External Magnetic Field—The Adiabatic Approximation.- XXII.4 A Spinning Quantum System in a Processing External Magnetic Field—The General Cyclic Evolution.- Problems.- XXIII A Quantum Physical System in a Quantum Environment—The Gauge Theory of Molecular Physics.- XXIII.1 Introduction.- XXIII.2 The Hamiltonian of the Diatomic Molecule.- XXIII.3 The Born-Oppenheimer Method.- XXIII.4 Gauge Theories.- XXIII.5 The Gauge Theory of Molecular Physics.- XXIII.6 The Electronic States of Diatomic Molecules.- XXIII.7 The Monopole of the Diatomic Molecule.- Problems.- Epilogue.