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Quantum Chemistry & Spectroscopy

Thomas Engel, Warren J. Hehre

Published Oct 17, 2009• 9 citations
OpenAlex

Abstract

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

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Morgan & Claypool Publishers eBooks•Feb 1, 2018•4 citations
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
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
Table of Contents 1. The Origins of Quantum Mechanics 1.1 Introduction 1.2 Blackbody Radiation The Problem with Blackbody Radiation 1.3 The Nature of Light The Photoelectric Effect The Compton Effect Is it a Particle or a Wave? 1.4 TheWave Nature of Matter 1.5 The Bohr Atom 1.6 Where do we Stand? 2. Math Interlude A: Complex Numbers and Linear Operators 2.1 Complex Numbers 2.2 Operators Definition of an Operator Eigenfunctions and Eigenvalues 3. The Schrodinger Equation 3.1 Derivation of the Schrodinger Equation 3.2 The Meaning of theWave Function 3.3 The Time-Independent Schrodinger Equation Derivation of the Time-Independent Schrodinger Equation Qualitative Solutions and the Origin of Quantization 4. One-Dimensional Time-Independent Schrodinger Equation 4.1 Unbound States: Scattering and Tunneling Scattering From Step-Function Potentials 4.2 Bound Systems The Infinite SquareWell The Harmonic Oscillator Potential 5. Math Interlude B: Linear Algebra 5.1 Properties of Linear Operators 5.2 Vector Spaces Inner Products Adjoint and Hermitian Operators Basis Sets 6. The Three-Dimensional Time-Independent Schrodinger Equation 6.1 Solution in Rectangular Coordinates 6.2 Angular Momentum 6.3 The Schrodinger Equation in Spherical Coordinates 6.4 The Hydrogen Atom 7. Math Interlude C: Matrices, Dirac Notation, and the Dirac Delta Function 7.1 The Matrix Formulation of Linear Operators 7.2 Dirac Notation 7.3 The Dirac Delta Function 8. Spin Angular Momentum 8.1 Spin Operators 8.2 Evidence for Spin 8.3 Adding Angular Momentum 8.4 The Matrix Representation of Spin 8.5 The Stern-Gerlach Experiment 8.6 Spin Precession 8.7 Spin Systems with Two Particles Noninteracting Spins Interacting Spins 8.8 Measurement Theory Hidden Variables The ManyWorlds Interpretation of Quantum Mechanics 9. Time-Independent Perturbation Theory 9.1 Derivation of Time-Independent Perturbation Theory 9.2 Perturbations to the Atomic Energy Levels Fine Structure The Lamb Shift 9.3 The Atom in External Electric or Magnetic Fields The Atom in an Electric Field: The Stark Effect The Atom in a Magnetic Field: The Zeeman Effect 10. The Variational Principle 10.1 Variational Principle: Theory 10.2 Variational Principle: Application to the Helium Atom 11. Time-Dependent Perturbation Theory 11.1 Derivation of Time-Dependent Perturbation Theory 11.2 Application: Selection Rules for Electromagnetic Radiation 12. Scattering Theory 12.1 Definition of the Cross Section 12.2 The Born Approximation 12.3 PartialWaves 13. The Multiparticle Schrodinger Equation 13.1 Wave Function for Identical Particles 13.2 Multielectron Atoms 14. Some Modern Applications of Quantum Mechanics 14.1 Magnetic Resonance Imaging 14.2 Quantum Computing 15. What Comes Next? Relativistic Quantum Mechanics 15.1 The Klein-Gordon Equation Derivation of the Klein-Gordon Equation Probability Densities and Currents 15.2 The Dirac Equation Answers and Hints for Selected End-of-Chapter Exercises Index
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel Kramers Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.