Mathematical Physical Chemistry: Practical and Intuitive Methodology

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
			3.7.2 Normalization of Radial Wave Functions
			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
			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
		Reference
	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
		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
Index