Abstract In this chapter, the mathematical machinery of quantum mechanics is further developed in order to address real-world 3-dimensional physics. 3-dimensional vector notation is used for quantum mechanical operators and the Schrödinger equation is presented in this notation. The density of states of a particle in a box is considered. The angular momentum operators are defined. The eigenfunctions of the Laplacian are found. The Schrödinger equation with a spherical potential is analysed and solved for a Coulomb potential. The spectroscopy of the hydrogen atom is discussed. The spin operators are introduced. The Stern–Gerlach experiment and the Zeeman effect are discussed. The quantum mechanics of identical particles is considered and fundamental particles are shown to behave as either bosons or fermions depending on their spin. The action and the Feynman path integral are shown to offer an alternative approach to quantum mechanics that elucidates the connection between quantum and classical physics.
Abstract In this chapter, the mathematical machinery of quantum mechanics is further developed in order to address real-world 3-dimensional physics. 3-dimensional vector notation is used for quantum mechanical operators and the Schrödinger equation is presented in this notation. The density of states of a particle in a box is considered. The angular momentum operators are defined. The eigenfunctions of the Laplacian are found. The Schrödinger equation with a spherical potential is analysed and solved for a Coulomb potential. The spectroscopy of the hydrogen atom is discussed. The spin operators are introduced. The Stern–Gerlach experiment and the Zeeman effect are discussed. The quantum mechanics of identical particles is considered and fundamental particles are shown to behave as either bosons or fermions depending on their spin. The action and the Feynman path integral are shown to offer an alternative approach to quantum mechanics that elucidates the connection between quantum and classical physics.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
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
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.