Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.
Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.
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.
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
Abstract The present work is a discussion of a number of simple problems of the free motion of electrons and atoms from the point of view of the wave mechanics. The author has had the advantage of many conversations with Prof. N. Bohr on the subject, and it was composed under this inspiration. Bohr has recently published his views on the foundations of the theory, and it would be out of place to enter here deeply into the matter, but some review of principles is unavoidable if the technical processes are to be understood. Perhaps the chief point of the work is to show how a simple, even old-fashioned, technique is entirely adequate to deal with these very new problems. 1. General Principles. The matrix and the wave mechanics have both been already developed to great lengths as a calculus of stationary states, but they have not yet got so far in what we may call dynamics, a description of the progress of events. More and more complicated phenomena have been fitted into the same scheme, but not much has been done in making this scheme intuitively understandable. One of the most important contributions is a recent work by Heisenberg, who develops an “uncertainty relation” by showing how each observation of a system, say an electron, always itself introduces some disturbance, so that the exact state at the beginning of each new experiment is essentially unknowable. Heisenberg works from the matrix point of view, but Bohr has pointed out that the “uncertainty relation,” exhibited by Heisenberg simply as an experimental result, becomes quite natural on wave principles, and is indeed nothing more than an expression of those principles.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.