Mathematical Physical Chemistry: Practical and Intuitive Methodology

Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Part I: Quantum Mechanics
Chapter 1: Schrödinger Equation and Its Application
1.1 Early-Stage Quantum Theory
1.2 Schrödinger Equation
1.3 Simple Applications of Schrödinger Equation
1.4 Quantum-Mechanical Operators and Matrices
1.5 Commutator and Canonical Commutation Relation
Reference
Chapter 2: Quantum-Mechanical Harmonic Oscillator
2.1 Classical Harmonic Oscillator
2.2 Formulation Based on an Operator Method
2.3 Matrix Representation of Physical Quantities
2.4 Coordinate Representation of Schrödinger Equation
2.5 Variance and Uncertainty Principle
References
Chapter 3: Hydrogen-Like Atoms
3.1 Introductory Remarks
3.2 Constitution of Hamiltonian
3.3 Separation of Variables
3.4 Generalized Angular Momentum
3.5 Orbital Angular Momentum: Operator Approach
3.6 Orbital Angular Momentum: Analytic Approach
3.6.1 Spherical Surface Harmonics and Associated Legendre Differential Equation
3.6.2 Orthogonality of Associated Legendre Functions
3.7 Radial Wave Functions of Hydrogen-Like Atoms
3.7.1 Operator Approach to Radial Wave Functions [3]
3.7.2 Normalization of Radial Wave Functions [10]
3.7.3 Associated Laguerre Polynomials
3.8 Total Wave Functions
References
Chapter 4: Optical Transition and Selection Rules
4.1 Electric Dipole Transition
4.2 One-Dimensional System
4.3 Three-Dimensional System
4.4 Selection Rules
4.5 Angular Momentum of Radiation [6]
References
Chapter 5: Approximation Methods of Quantum Mechanics
5.1 Perturbation Method
5.1.1 Quantum State and Energy Level Shift Caused by Perturbation
5.1.2 Several Examples
5.2 Variational Method
References
Chapter 6: Theory of Analytic Functions
6.1 Set and Topology
6.1.1 Basic Notions and Notations
6.1.2 Topological Spaces and Their Building Blocks
(a) Neighborhoods [4]
(b) Interior and Closure [4]
(c) Boundary [4]
(d) Accumulation Points and Isolated Points
(e) Connectedness
6.1.3 T1-Space
6.1.4 Complex Numbers and Complex Plane
6.2 Analytic Functions of a Complex Variable
6.3 Integration of Analytic Functions: Cauchy´s Integral Formula
6.4 Taylor´s Series and Laurent´s Series
6.5 Zeros and Singular Points
6.6 Analytic Continuation
6.7 Calculus of Residues
6.8 Examples of Real Definite Integrals
6.9 Multivalued Functions and Riemann Surfaces
6.9.1 Brief Outline
6.9.2 Examples of Multivalued Functions
References
Part II: Electromagnetism
Chapter 7: Maxwell´s Equations
7.1 Maxwell´s Equations and Their Characteristics
7.2 Equation of Wave Motion
7.3 Polarized Characteristics of Electromagnetic Waves
7.4 Superposition of Two Electromagnetic Waves
References
Chapter 8: Reflection and Transmission of Electromagnetic Waves in Dielectric Media
8.1 Electromagnetic Fields at an Interface
8.2 Basic Concepts Underlying Phenomena
8.3 Transverse Electric (TE) Waves and Transverse Magnetic (TM) Waves
8.4 Energy Transport by Electromagnetic Waves
8.5 Brewster Angles and Critical Angles
8.6 Total Reflection
8.7 Waveguide Applications
8.7.1 TE and TM Waves in a Waveguide
8.7.2 Total Internal Reflection and Evanescent Waves
8.8 Stationary Waves
References
Chapter 9: Light Quanta: Radiation and Absorption
9.1 Blackbody Radiation
9.2 Planck´s Law of Radiation and Mode Density of Electromagnetic Waves
9.3 Two-Level Atoms
9.4 Dipole Radiation
9.5 Lasers
9.5.1 Brief Outlook
9.5.2 Organic Lasers
9.6 Mechanical System
References
Chapter 10: Introductory Green´s Functions
10.1 Second-Order Linear Differential Equations (SOLDEs)
10.2 First-Order Linear Differential Equations (FOLDEs)
10.3 Second-Order Differential Operators
10.4 Green´s Functions
10.5 Construction of Green´s Functions
10.6 Initial-Value Problems (IVPs)
10.6.1 General Remarks
10.6.2 Green´s Functions for IVPs
10.6.3 Estimation of Surface Terms
10.6.4 Examples
10.7 Eigenvalue Problems
References
Part III: Linear Vector Spaces
Chapter 11: Vectors and Their Transformation
11.1 Vectors
11.2 Linear Transformations of Vectors
11.3 Inverse Matrices and Determinants
11.4 Basis Vectors and Their Transformations
References
Chapter 12: Canonical Forms of Matrices
12.1 Eigenvalues and Eigenvectors
12.2 Eigenspaces and Invariant Subspaces
12.3 Generalized Eigenvectors and Nilpotent Matrices
12.4 Idempotent Matrices and Generalized Eigenspaces
12.5 Decomposition of Matrix
12.6 Jordan Canonical Form
12.6.1 Canonical Form of Nilpotent Matrix
12.6.2 Jordan Blocks
12.6.3 Example of Jordan Canonical Form
12.7 Diagonalizable Matrices
References
Chapter 13: Inner Product Space
13.1 Inner Product and Metric
13.2 Gram Matrices
13.3 Adjoint Operators
13.4 Orthonormal Basis
References
Chapter 14: Hermitian Operators and Unitary Operators
14.1 Projection Operators
14.2 Normal Operators
14.3 Unitary Diagonalization of Matrices
14.4 Hermitian Matrices and Unitary Matrices
14.5 Hermitian Quadratic Forms
14.6 Simultaneous Eigenstates and Diagonalization
References
Chapter 15: Exponential Functions of Matrices
15.1 Functions of Matrices
15.2 Exponential Functions of Matrices and Their Manipulations
15.3 System of Differential Equations
15.3.1 Introduction
15.3.2 System of Differential Equations in a Matrix Form: Resolvent Matrix
15.3.3 Several Examples
15.4 Motion of a Charged Particle in Polarized Electromagnetic Wave
References
Part IV: Group Theory and Its Chemical Applications
Chapter 16: Introductory Group Theory
16.1 Definition of Groups
16.2 Subgroups
16.3 Classes
16.4 Isomorphism and Homomorphism
16.5 Direct-Product Groups
Reference
Chapter 17: Symmetry Groups
17.1 A Variety of Symmetry Operations
17.2 Successive Symmetry Operations
17.3 O and Td Groups
17.4 Special Orthogonal Group SO(3)
17.4.1 Rotation Axis and Rotation Matrix
17.4.2 Euler Angles and Related Topics
References
Chapter 18: Representation Theory of Groups
18.1 Definition of Representation
18.2 Basis Functions of Representation
18.3 Schur´s Lemmas and Grand Orthogonality Theorem (GOT)
18.4 Characters
18.5 Regular Representation and Group Algebra
18.6 Classes and Irreducible Representations
18.7 Projection Operators: Revisited
18.8 Direct-Product Representation
18.9 Symmetric Representation and Antisymmetric Representation
References
Chapter 19: Applications of Group Theory to Physical Chemistry
19.1 Transformation of Functions
19.2 Method of Molecular Orbitals (MOs)
19.3 Calculation Procedures of Molecular Orbitals (MOs)
19.4 MO Calculations Based on π-Electron Approximation
19.4.1 Ethylene
19.4.2 Cyclopropenyl Radical [1]
19.4.3 Benzene
19.4.4 Allyl Radical [1]
19.5 MO Calculations of Methane
References
Chapter 20: Theory of Continuous Groups
20.1 Introduction: Operators of Rotation and Infinitesimal Rotation
20.2 Rotation Groups: SU(2) and SO(3)
20.2.1 Construction of SU(2) Matrices
20.2.2 SU(2) Representation Matrices: Wigner Formula
20.2.3 SO(3) Representation Matrices and Spherical Surface Harmonics
20.2.4 Irreducible Representations of SU(2) and SO(3)
20.2.5 Parameter Space of SO(3)
20.2.6 Irreducible Characters of SO(3) and Their Orthogonality
20.3 Clebsch-Gordan Coefficients of Rotation Groups
20.3.1 Direct-Product of SU(2) and Clebsch-Gordan Coefficients
20.3.2 Calculation Procedures of Clebsch-Gordan Coefficients
20.3.3 Examples of Calculation of Clebsch-Gordan Coefficients
20.4 Lie Groups and Lie Algebras
20.4.1 Definition of Lie Groups and Lie Algebras: One-Parameter Groups
20.4.2 Properties of Lie Algebras
20.4.3 Adjoint Representation of Lie Groups
20.5 Connectedness of Lie Groups
20.5.1 Several Definitions and Examples
20.5.2 O(3) and SO(3)
20.5.3 Simply Connected Lie Groups: Local Properties and Global Properties
References
Part V: Introduction to the Quantum Theory of Fields
Chapter 21: The Dirac Equation
21.1 Historical Background
21.2 Several Remarks on the Special Theory of Relativity
21.2.1 Minkowski Space and Lorentz Transformation
21.2.2 Event and World Interval
21.3 Constitution and Solutions of the Dirac Equation
21.3.1 General Form of the Dirac Equation
21.3.2 Plane Wave Solutions of the Dirac Equation
21.3.3 Negative-Energy Solution of the Dirac Equation
21.3.4 One-Particle Hamiltonian of the Dirac Equation
21.4 Normalization of the Solutions of the Dirac Equation
21.5 Charge Conjugation
21.6 Characteristics of the Gamma Matrices
References
Chapter 22: Quantization of Fields
22.1 Lagrangian Formalism of the Fields [1]
22.2 Introductory Fourier Analysis [3]
22.2.1 Fourier Series Expansion
22.2.2 Fourier Integral Transforms: Fourier Transform and Inverse Fourier Transform
22.3 Quantization of the Scalar Field [5, 6]
22.3.1 Lagrangian Density and Action Integral
22.3.2 Equal-Time Commutation Relation and Field Quantization
22.3.3 Hamiltonian and Fock Space
22.3.4 Invariant Delta Functions of the Scalar Field
22.3.5 Feynman Propagator of the Scalar Field
22.3.6 General Consideration on the Field Quantization
22.4 Quantization of the Dirac Field [5, 6]
22.4.1 Lagrangian Density and Hamiltonian Density of the Dirac Field
22.4.2 Quantization Procedure of the Dirac Field
22.4.3 Antiparticle: Positron
22.4.4 Invariant Delta Functions of the Dirac Field
22.4.5 Feynman Propagator of the Dirac Field
22.5 Quantization of the Electromagnetic Field [5-7]
22.5.1 Relativistic Formulation of the Electromagnetic Field
22.5.2 Lagrangian Density and Hamiltonian Density of the Electromagnetic Field
22.5.3 Polarization Vectors of the Electromagnetic Field [5]
22.5.4 Canonical Quantization of the Electromagnetic Field
22.5.5 Hamiltonian and Indefinite Metric
22.5.6 Feynman Propagator of Photon Field
References
Chapter 23: Interaction Between Fields
23.1 Lorentz Force and Minimal Coupling
23.2 Lagrangian and Field Equation of the Interacting Fields
23.3 Local Phase Transformation and U(1) Gauge Field
23.4 Interaction Picture
23.5 S-Matrix and S-Matrix Expansion
23.6 N-Product and T-Product
23.6.1 Example of Two Field Operators
23.6.2 Calculations Including Both the N-Products and T-Products
23.7 Feynman Rules and Feynman Diagrams in QED
23.7.1 Zeroth- and First-Order S-Matrix Elements
23.7.2 Second-Order S-Matrix Elements and Feynman Diagrams
23.8 Example: Compton Scattering [1]
23.8.1 Introductory Remarks
23.8.2 Feynman Amplitude and Feynman Diagrams of Compton Scattering
23.8.3 Scattering Cross-Section [1]
23.8.4 Spin and Photon Polarization Sums
23.8.5 Detailed Calculation Procedures of Feynman Amplitude [1]
23.8.6 Experimental Observations
23.9 Summary
References
Chapter 24: Basic Formalism
24.1 Extended Concepts of Vector Spaces
24.1.1 Bilinear Mapping
24.1.2 Tensor Product
24.1.3 Bilinear Form
24.1.4 Dual Vector Space
24.1.5 Invariants
24.1.6 Tensor Space and Tensors
24.1.7 Euclidean Space and Minkowski Space
24.2 Lorentz Group and Lorentz Transformations
24.2.1 Lie Algebra of the Lorentz Group
24.2.2 Successive Lorentz Transformations [9]
24.3 Covariant Properties of the Dirac Equation
24.3.1 General Consideration on the Physical Equation
24.3.2 The Klein-Gordon Equation and the Dirac Equation
24.4 Determination of General Form of Matrix S(Λ)
24.5 Transformation Properties of the Dirac Spinors [9]
24.6 Transformation Properties of the Dirac Operators [9]
24.6.1 Case I: Single Lorentz Boost
24.6.2 Case II: Non-Coaxial Lorentz Boosts
24.7 Projection Operators and Related Operators for the Dirac Equation
24.8 Spectral Decomposition of the Dirac Operators
24.9 Connectedness of the Lorentz Group
24.9.1 Polar Decomposition of a Non-Singular Matrix [14, 15]
24.9.2 Special Linear Group: SL(2,)
24.10 Representation of the Proper Orthochronous Lorentz Group SO0(3,1) [6, 8]
References
Chapter 25: Advanced Topics of Lie Algebra
25.1 Differential Representation of Lie Algebra [1]
25.1.1 Overview
25.1.2 Adjoint Representation of Lie Algebra [1]
25.1.3 Differential Representation of Lorentz Algebra [1]
25.2 Cartan-Weyl Basis of Lie Algebra [1]
25.2.1 Complexification
25.2.2 Coupling of Angular Momenta: Revisited
25.3 Decomposition of Lie Algebra
25.4 Further Topics of Lie Algebra
25.5 Closing Remarks
References
Index