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Elementary Quantum Mechanics

Ronald W. Gurney

Published Nov 1, 2015• 2 citations
OpenAlex

Abstract

Preface 1. Energy diagrams 2. The new language of physics and chemistry 3. The wave equation - the hydrogen atom - magnetic moment and electron spin - simple problems 4. The uncertainty principle - systems containing many particles - electron configurations and the periodic table - metals 5. The movement of particles 6. Two interacting particles - diatomic molecules 7. Independent observable quantities - homonuclear molecules - valence bonds 8. Electrons in crystals - insulators and conductors 9. Perturbation theory - the description of physical events 10. The same continued Mathematical appendix Table of electron configurations and first ionisation potentials Subject index Name index.

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•Oct 17, 2009•9 citations
Preface 1. Energy diagrams 2. The new language of physics and chemistry 3. The wave equation - the hydrogen atom - magnetic moment and electron spin - simple problems 4. The uncertainty principle - systems containing many particles - electron configurations and the periodic table - metals 5. The movement of particles 6. Two interacting particles - diatomic molecules 7. Independent observable quantities - homonuclear molecules - valence bonds 8. Electrons in crystals - insulators and conductors 9. Perturbation theory - the description of physical events 10. The same continued Mathematical appendix Table of electron configurations and first ionisation potentials Subject index Name index.
PREFACE PART I: THE STRUCTURE OF MATTER 1. THE MICROSCOPIC WORLD: ATOMS AND MOLECULES 1.1 Development of the Atomic Theory: Relative Atomic Weights 1.2 Atomic Magnitudes 1.3 The Charge-to-Mass Ratio of the Electron: Thomson's Method 1.4 The Charge of the Electron: Millikan's Method 1.5 Mass Spectrometry 1.6 The Atomic Mass Scale and the Mole 1.7 The Periodic Table 2. ORIGINS OF THE QUANTUM THEORY OF MATTER 2.1 The Franck-Hertz Experiment 2.2 The Photoelectric Effect 2.3 X-rays and Matter 2.4 The Emission Spectra of Atoms 2.5 The Nuclear Atom 2.6 The Problem of Black-Body Radiation 2.7 The Concept of Action 2.8 The Harmonic Oscillator 2.9 Action Quantized: The Heat Capacity of Solids 2.10 Some Orders of Magnitude 2.11 Bohr's Model of the Atom Appendix 2A: Rutherford Scattering 3. MATTER WAVES IN SIMPLE SYSTEMS 3.1 The de Broglie Hypothesis 3.2 The Nature of Waves 3.3 Dispersion Relations and Wave Equations: The Free Particle 3.4 Operators 3.5 Eigenfunctions and Eigenvalues 3.6 The Particle in a One-Dimensional Box 3.7 The Indeterminacy or Uncertainty Principle 3.8 Expectation Values Summary of Postulates 3.9 Particles in Two- and Three-Dimensional Boxes 3.10 Particles in Circular Boxes 3.11 Particles in Spherical Boxes 3.12 The Rigid Rotor Appendix 3A: More on Circular Coordinates and the Circular Box 4. PARTICLES IN VARYING POTENTIAL FIELDS TRANSITIONS 4.1 Finite Potential Barriers 4.2 The Quantum Mechanical Harmonic Oscillator 4.3 The Hydrogen Atom 4.4 The Shapes of Orbitals 4.5 Transitions between Energy Levels 5. THE STRUCTURE OF ATOMS 5.1 Electron Spin Magnetic Phenomena 5.2 The Pauli Exclusion Principle the Aufbau Principle 5.3 Electronic Configurations of Atoms 5.4 Calculation of Atomic Structures 5.5 Atomic Structure and Periodic Behavior 5.6 Term Splitting and the Vector Model 5.7 Fine Structure and Spin-Orbit Interactions Appendix 5A: The Stern-Gerlach Experiment 6. THE CHEMICAL BOND IN THE SIMPLEST MOLECULES: H[2+ AND H[2 6.1 Bonding Forces between Atoms 6.2 The Simplest Molecule: The Hydrogen Molecule-Ion, H[2+ 6.3 H[2+: Molecular Orbitals and the LCAO Approximation 6.4 H[2+: Obtaining the Energy Curve 6.5 H[2+: Correlation of Orbitals Excited States 6.6 The H[2 Molecule: Simple MO Description 6.7 Symmetry Properties of Identical Particles 6.8 H[2: The Valence Bond Representation 6.9 H[2: Beyond the Simple MO and VB Approximations 6.10 H[2: Excited Electronic States Appendix 6A: Orthogonality Appendix 6B: Hermitian Operators 7. MORE ABOUT DIATOMIC MOLECULES 7.1 Vibrations of Diatomic Molecules 7.2 Rotations of Diatomic Molecules 7.3 Spectra of Diatomic Molecules 7.4 The Ionic Bond 7.5 Homonuclear Diatomic Molecules: Molecular Orbitals and Orbital Correlation 7.6 Homonuclear Diatomic Molecules: Aufbau Principle and Structure of First-Row Molecules 7.7 Introduction to Heteronuclear Diatomic Molecules: Electronegativity 7.8 Bonding in LiH: Crossing and Noncrossing Potential Curves 7.9 Other First-Row Diatomic Hydrides 7.10 Isoelectronic and Other Series Appendix 7A: Perturbation Theory 8. TRIATOMIC MOLECULES 8.1 Electronic Structure and Geometry in the Simplest Cases: H[3 and H[3+ 8.2 Dihydrides: Introduction to the Water Molecule 8.3 Hybrid Orbitals 8.4 Delocalized Orbitals in H[2O: The General MO Method 8.5 Bonding in More Complex Triatomic Molecules 8.6 Normal Coordinates and Modes of Vibration 8.7 A Solvable Example: The Vibrational Modes of CO[2 8.8 Transitions and Spectra of Polyatomic Molecules: Rotations and Vibrations 8.9 Transitions and Spectra of Polyatomic Molecules: Magnetic Transitions 8.10 Transitions and Spectra of Polyatomic Molecules: Electronic Transitions 9. LARGER POLYATOMIC MOLECULES 9.1 Small Molecules 9.2 Catenated Carbon Compounds Transferability 9.3 Other Extended Structures 9.4 Some Steric Effects 9.5 Complex Ions and Other Coordination Compounds: Simple Polyhedra 9.6 Chirality and Optical Rotation 9.7 Chiral and Other Complex Ions 9.8 Magnetic Properties of Complexes 9.9 Electronic Structure of Complexes Appendix 9A: Schmidt Orthogonalization 10. INTERMOLECULAR FORCES 10.1 Long-Range Forces: Interactions between Charge Distributions 10.2 Empirical Intermolecular Potentials 10.3 Weakly Associated Molecules 11. THE STRUCTURE OF SOLIDS 11.1 Some General Properties of Solids 11.2 Space Lattices and Crystal Symmetry 11.3 X-ray Diffraction from Crystals: The Bragg Model 11.4 The Laue Model 11.5 Determination of Crystal Structures 11.6 Techniques of Diffraction 11.7 Molecular Crystals 11.8 Structures of Ionic Crystals 11.9 Binding Energy of Ionic Crystals 11.10 Covalent Solids 11.11 The Free-Electron Theory of Metals 11.12 The Band Theory of Solids 11.13 Conductors, Insulators, and Semiconductors 11.14 Other Forms of Condensed Matter APPENDICES I. Systems of Units II. Partial Derivatives III. Glossary of Symbols IV. Searching the Scientific Literature INDEX
I. RELATIVITY. 1. The Space and Time of Relativity. 2. Relativistic Mechanics. II. QUANTUM MECHANICS. 3. Atoms. 4. Quantization of Light. 5. Quantization of Atomic Energy Levels. 6. Matter Waves. 7. The Schrodinger Equation in One Dimension. 8. The Three-Dimensional Schrodinger Equation. 9. Electron Spin. 10. Multielectron Atoms the Pauli Principle and the Periodic Table. 11. Atomic Transitions and Radiation. III. SYSTEMS WITH TWO OR MORE ATOMS. 12. Molecules. 13. Solids - Theory. 14. Solids - Applications. 15. Statistical Mechanics. IV. SUBATOMIC PHYSICS. 16. The Structure of Atomic Nuclei. 17. Radioactivity and Nuclear Reactions. 18. Elementary Particles. APPENDIXES. A. Physical Constants. B. Useful Mathematical Relations. C. Alphabetical List of the Elements. D. Atomic and Nuclear Data. Suggestions for Further Reading. Picture Credits and References. Answers to Odd-Numbered Problems. Index.
Preface * Part I. Introductory Concepts in Quantum Theory * Units of Measurement * Quantum Mechanics---Some Remarks and Themes * Equations of Motion in Classical Mechanics * Quantization and the Schrodinger Equation * Hypergeometric Equations and Special Functions * Hydrogen-like Atoms * Heisenberg's Uncertainty Principle * Group Representations and Selection Rules * A Charged Particle in an Electro-magnetic Field * Spin Wave Functions * Introduction to Multi-electron Atoms * Part II. Some Selected Topics * Fresnel Integrals and Feynman Integrals * Path Integral for the Harmonic Oscillator * Euclidean Path Integrals * The Density Matrix and Partition Function * Zeta Regularization * Helmholtz Free Energy and the Selberg Trace Formula * The Zeta Function of a Product * Two Guest Lectures * Some Further Electron Configurations * Mendeleev Periodic Table * Another Example of Zeta Regularization * Integral Evaluation * Some Informal Comments on QFT * References * Index
The textbook presents the physical theory of quantum mechanics. Contents: principles of quantum mechanics, energy and impulse, the Schroedinger equation, the angular momentum, movement in a radial symmetric field, perturbation theory, the quasi-classic case, the spin, identical particles, the atom, the two-atomic molecule, the theory of symmetry, multi-atomic molecules, the addition of angular momenta, movement in the magnetic field, structure of the nucleus, elastic collisions, inelastic collisions, and mathematical supplements. A subject index is included.
CHAPTER 1: FROM CLASSICAL TO QUANTUM MECHANICS 1.1 Why Study Quantum Mechanics? 1.2 Quantum Mechanics Arose Out of the Interplay of Experiments and Theory 1.3 Blackbody Radiation 1.4 The Photoelectric Effect 1.5 Particles Exhibit Wave-Like Behavior 1.6 Diffraction by a Double Slit 1.7 Atomic Spectra and the Bohr Model for the Hydrogen Atom CHAPTER 2: THE SCHRODINGER EQUATION 2.1 What Determines If a System Needs to Be Described Using Quantum Mechanics? 2.2 Classical Waves and the Nondispersive Wave Equation 2.3 Waves Are Conveniently Represented as Complex Functions 2.4 Quantum Mechanical Waves and the Schrodinger Equation 2.5 Solving the Schrodinger Equation: Operators, Observables, Eigenfunctions, and Eigenvalues 2.6 The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal 2.7 The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set 2.8 Summing Up the New Concepts CHAPTER 3: THE QUANTUM MECHANICAL POSTULATES 3.1 The Physical Meaning Associated with the Wave Function 3.2 Every Observable Has a Corresponding Operator 3.3 The Result of an Individual Measurement 3.4 The Expectation Value 3.5 The Evolution in Time of a Quantum Mechanical System CHAPTER 4: USING QUANTUM MECHANICS ON SIMPLE SYSTEMS 4.1 The Free Particle 4.2 The Particle in a One-Dimensional Box 4.3 Two- and Three-Dimensional Boxes 4.4 Using the Postulates to Understand the Particle in the Box and Vice Versa CHAPTER 5: THE PARTICLE IN THE BOX AND THE REAL WORLD 5.1 The Particle in the Finite Depth Box 5.2 Differences in Overlap between Core and Valence Electrons 5.3 Pi Electrons in Conjugated Molecules Can Be Treated as Moving Freely in a Box 5.4 Why Does Sodium Conduct Electricity and Why Is Diamond an Insulator? 5.5 Tunneling through a Barrier 5.6 The Scanning Tunneling Microscope 5.7 Tunneling in Chemical Reactions 5.8 (Supplemental) Quantum Wells and Quantum Dots CHAPTER 6: COMMUTING AND NONCOMMUTING OPERATORS AND THE SURPRISING CONSEQUENCES OF ENTANGLEMENT 6.1 Commutation Relations 6.2 The Stern-Gerlach Experiment 6.3 The Heisenberg Uncertainty Principle 6.4 (Supplemental) The Heisenberg Uncertainty Principle Expressed in Terms of Standard Deviations 6.5 (Supplemental) A Thought Experiment Using a Particle in a Three-Dimensional Box 6.6 (Supplemental) Entangled States, Teleportation, and Quantum Computers CHAPTER 7: A QUANTUM MECHANICAL MODEL FOR THE VIBRATION AND ROTATION OF MOLECULES 7.1 Solving the Schrodinger Equation for the Quantum Mechanical Harmonic Oscillator 7.2 Solving the Schrodinger Equation for Rotation in Two Dimensions 7.3 Solving the Schrodinger Equation for Rotation in Three Dimensions 7.4 The Quantization of Angular Momentum 7.5 The Spherical Harmonic Functions 7.6 (Optional Review) The Classical Harmonic Oscillator 7.7 (Optional Review) Angular Motion and the Classical Rigid Rotor 7.8 (Supplemental) Spatial Quantization CHAPTER 8: THE VIBRATIONAL AND ROTATIONAL SPECTROSCOPY OF DIATOMIC MOLECULES 8.1 An Introduction to Spectroscopy 8.2 Absorption, Spontaneous Emission, and Stimulated Emission 8.3 An Introduction to Vibrational Spectroscopy 8.4 The Origin of Selection Rules 8.5 Infrared Absorption Spectroscopy 8.6 Rotational Spectroscopy 8.7 (Supplemental) Fourier Transform Infrared Spectroscopy 8.8 (Supplemental) Raman Spectroscopy 8.9 (Supplemental) How Does the Transition Rate between States Depend on Frequency? CHAPTER 9: THE HYDROGEN ATOM 9.1 Formulating the Schrodinger Equation 9.2 Solving the Schrodinger Equation for the Hydrogen Atom 9.3 Eigenvalues and Eigenfunctions for the Total Energy 9.4 The Hydrogen Atom Orbitals 9.5 The Radial Probability Distribution Function 9.6 The Validity of the Shell Model of an Atom CHAPTER 10: MANY-ELECTRON ATOMS 10.1 Helium: The Smallest Many-Electron Atom 10.2 Introducing Electron Spin 10.3 Wave Functions Must Reflect the Indistinguishability of Electrons 10.4 Using the Variational Method to Solve the Schrodinger Equation 10.5 The Hartree-Fock Self-Consistent Field Method 10.6 Understanding Trends in the Periodic Table from Hartree-Fock Calculations CHAPTER 11: QUANTUM STATES FOR MANY-ELECTRON ATOMS AND ATOMIC SPECTROSCOPY 11.1 Good Quantum Numbers, Terms, Levels, and States 11.2 The Energy of a Configuration Depends on Both Orbital and Spin Angular Momentum 11.3 Spin-Orbit Coupling Breaks Up a Term into Levels 11.4 The Essentials of Atomic Spectroscopy 11.5 Analytical Techniques Based on Atomic Spectroscopy 11.6 The Doppler Effect 11.7 The Helium-Neon Laser 11.8 Laser Isotope Separation 11.9 Auger Electron and X-Ray Photoelectron Spectroscopies 11.10 Selective Chemistry of Excited States: O(3P) and O(1D) 11.11 (Supplemental) Configurations with Paired and Unpaired Electron Spins Differ in Energy CHAPTER 12: THE CHEMICAL BOND IN DIATOMIC MOLECULES 12.1 The Simplest One-Electron Molecule: 12.2 The Molecular Wave Function for Ground-State 12.3 The Energy Corresponding to the H2+ Molecular Wave Functions 12.4 A Closer Look at the H2+ Molecular Wave Functions 12.5 Combining Atomic Orbitals to Form Molecular Orbitals 12.6 Molecular Orbitals for Homonuclear Diatomic Molecules 12.7 The Electronic Structure of Many-Electron Molecules 12.8 Bond Order, Bond Energy, and Bond Length 12.9 Heteronuclear Diatomic Molecules 12.10 The Molecular Electrostatic Potential CHAPTER 13: MOLECULAR STRUCTURE AND ENERGY LEVELS FOR POLYATOMIC MOLECULES 13.1 Lewis Structures and the VSEPR Model 13.2 Describing Localized Bonds Using Hybridization for Methane, Ethene, and Ethyne 13.3 Constructing Hybrid Orbitals for Nonequivalent Ligands 13.4 Using Hybridization to Describe Chemical Bonding 13.5 Predicting Molecular Structure Using Molecular Orbital Theory 13.6 How Different Are Localized and Delocalized Bonding Models? 13.7 Qualitative Molecular Orbital Theory for Conjugated and Aromatic Molecules: The Huckel Model 13.8 From Molecules to Solids 13.9 Making Semiconductors Conductive at Room Temperature CHAPTER 14: ELECTRONIC SPECTROSCOPY 14.1 The Energy of Electronic Transitions 14.2 Molecular Term Symbols 14.3 Transitions between Electronic States of Diatomic Molecules 14.4 The Vibrational Fine Structure of Electronic Transitions in Diatomic Molecules 14.5 UV-Visible Light Absorption in Polyatomic Molecules 14.6 Transitions among the Ground and Excited States 14.7 Singlet--Singlet Transitions: Absorption and Fluorescence 14.8 Intersystem Crossing and Phosphorescence 14.9 Fluorescence Spectroscopy and Analytical Chemistry 14.10 Ultraviolet Photoelectron Spectroscopy 14.11 Single Molecule Spectroscopy 14.12 Fluorescent Resonance Energy Transfer (FRET) 14.13 Linear and Circular Dichroism 14.14 (Supplemental) Assigning + and -- to Terms of Diatomic Molecules CHAPTER 15: COMPUTATIONAL CHEMISTRY 15.1 The Promise of Computational Chemistry 15.2 Potential Energy Surfaces 15.3 Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrodinger Equation 15.4 Properties of Limiting Hartree-Fock Models 15.5 Theoretical Models and Theoretical Model Chemistry 15.6 Moving Beyond Hartree-Fock Theory 15.7 Gaussian Basis Sets 15.8 Selection of a Theoretical Model 15.9 Graphical Models 15.10 Conclusion CHAPTER 16: MOLECULAR SYMMETRY 16.1 Symmetry Elements, Symmetry Operations, and Point Groups 16.2 Assigning Molecules to Point Groups 16.3 The H2O Molecule and the C2v Point Group 16.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table 16.5 The Dimension of a Representation 16.6 Using the C2v Representations to Construct Molecular Orbitals for H2O 16.7 The Symmetries of the Normal Modes of Vibration of Molecules 16.8 Selection Rules and Infrared versus Raman Activity 16.9 (Supplemental) Using the Projection Operator Method to Generate MOs That Are Bases for Irreducible Representations CHAPTER 17: NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY 17.1 Intrinsic Nuclear Angular Momentum and Magnetic Moment 17.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic Field 17.3 The Chemical Shift for an Isolated Atom 17.4 The Chemical Shift for an Atom Embedded in a Molecule 17.5 Electronegativity of Neighboring Groups and Chemical Shifts 17.6 Magnetic Fields of Neighboring Groups and Chemical Shifts 17.7 Multiplet Splitting of NMR Peaks Arises through Spin--Spin Coupling 17.8 Multiplet Splitting When More Than Two Spins Interact 17.9 Peak Widths in NMR Spectroscopy 17.10 Solid-State NMR 17.11 NMR Imaging 17.12 (Supplemental) The NMR Experiment in the Laboratory and Rotating Frames 17.13 (Supplemental) Fourier Transform NMR Spectroscopy 17.14 (Supplemental) Two-Dimensional NMR