An Introduction to Groups and their Matrices for Science Students

✍ Scribed by Robert Kolenkow

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 338
• Edition: New
• Category: Library

✦ Synopsis

Group theory, originating from algebraic structures in mathematics, has long been a powerful tool in many areas of physics, chemistry and other applied sciences, but it has seldom been covered in a manner accessible to undergraduates. This book from renowned educator Robert Kolenkow introduces group theory and its applications starting with simple ideas of symmetry, through quantum numbers, and working up to particle physics. It features clear explanations, accompanying problems and exercises, and numerous worked examples from experimental research in the physical sciences. Beginning with key concepts and necessary theorems, topics are introduced systematically including: molecular vibrations and lattice symmetries; matrix mechanics; wave mechanics; rotation and quantum angular momentum; atomic structure; and finally particle physics. This comprehensive primer on group theory is ideal for advanced undergraduate topics courses, reading groups, or self-study, and it will help prepare graduate students for higher-level courses.

✦ Table of Contents

Title page
Contents
Preface
1 Fundamental Concepts
1.1 Introduction
1.2 Operations
1.3 What Is a Group?
1.4 Examples of Groups
1.5 Matrix Representations of Groups
1.6 Matrix Algebra
1.7 Special Matrices
1.8 A Brief History of Group Theory
1.9 BriefBios
Summary of Chapter 1
Problems and Exercises
2 Matrix Representations of Discrete Groups
2.1 Introduction
2.2 Basis Functions and Representations
2.3 Similarity Transformations
2.4 Equivalent Representations
2.5 Similarity Transformations and Unitary Matrices
2.6 Character and Its Invariance under Similarity Transformations
2.7 Irreducible Representations
2.8 Kronecker (Direct) Product
2.9 Kronecker Sum
Summary of Chapter 2
Stated Theorems in Chapter 2
Problems and Exercises
3 Molecular Vibrations
3.1 Introduction
3.2 Oscillating Systems and Newton's Laws
3.3 Normal Modes and Group Theory
3.4 Normal Modes of a Water Molecule
3.5 Visualizing Normal Modes
3.6 Infrared (IR) Spectroscopy
3.7 Raman Spectroscopy
3.8 Brief Bios
Summary of Chapter 3
Problems and Exercises
4 Crystalline Solids
4.1 Introduction
4.2 Bravais Lattices
4.3 X Ray Crystallography
4.4 Fourier Transform
4.5 Reciprocal Lattice
4.6 Lattice Translation Group
4.7 Crystallographic Point Groups and Rotation Symmetry
4.8 Crystallographic Space Groups and the Seitz Operator
4.9 Crystal Symmetry Operations
4.10 Lattice Vibrations
4.11 BriefBios
Summary of Chapter 4
Problems and Exercises
5 Bohr's Quantum Theory and Matrix Mechanics
5.1 Introduction
5.2 Bohr's Model
5.3 Matrix Mechanics
5.4 Matrix Mechanics Quantization
5.5 Consequences of Matrix Mechanics
5.6 Heisenberg Uncertainty Relation
5.7 BriefBios
Summary of Chapter 5
Problems and Exercises
6 Wave Mechanics, Measurement, and Entanglement
6.1 Introduction
6.2 Schrodinger's Wave Mechanics
6.3 The Wave Equation
6.4 Quantization Conditions in Wave Mechanics
6.5 Matrix Diagonalization
6.6 Quantum Measurement
6.7 The EPR Paradox and Entanglement
6.8 BriefBio
Summary of Chapter 6
Problems and Exercises
7 Rotation
7.1 Introduction
7.2 Two Ways of Looking at Rotation
7.3 Rotation of a Function
7.4 The Axial Rotation Group
7.5 The U(l) and SU(2) Groups
7.6 Pauli Matrices, SU(2), and Rotation
7.7 Euler Angles
7.8 Finite Rotations Don't Commute
7.9 ... But Rotations Do Commute to First Order
7.10 BriefBios
Summary of Chapter 7
Problems and Exercises
8 Quantum Angular Momentum
8.1 Introduction
8.2 Stern and Gerlach: An Important Experiment (1922)
8.3 Rotation and Angular Momentum Operators
8.4 Commutation Relations
8.5 The Axial Rotation Group Again
8.6 Raising and Lowering (Ladder) Operators
8.7 Angular Momentum Operators and Representations of the Rotation Group
8.8 The Ujm Are Spherical Harmonics
8.9 Spin Basis Functions and Pauli Matrices
8.10 Coupling (Adding) Angular Momenta
8.11 Wigner-Eckart Theorem
8.12 Selection Rules
8.13 BriefBios
Summary of Chapter 8
Problems and Exercises
9 The Structure of Atoms
9.1 Introduction
9.2 Zeeman: An Important Experiment (1897)
9.3 Quantum Theory of the Zeeman Effect
9.4 Fine Structure
9.5 Example: Intermediate Field Zeeman
9.6 Nuclear Spin and Hyperfme Structure
9.7 Multi-electron Atoms
9.8 The Helium Atom
9.9 The Structure of Multi electron Atoms
Summary of Chapter 9
Problems and Exercises
10 Particle Physics
10.1 Introduction
10.2 Natural Units
10.3 Isospin
10.4 Cross Section
10.5 Antiparticles
10.6 The Lagrangian
10.7 Gauge Theory
10.8 All Those Particles - the Particle Zoo
10.9 The Quark Model
10.10 Conservation Laws and Quantum Numbers
10.11 Group Theory and Particle Physics
10.12 Concluding Remark
Summary of Chapter 10
Problems and Exercises
Appendix A Character Tables from Class Sums
Appendix  Born-Jordan Proof of the Quantization Condition
Appendix Ñ Weyl Derivation of the Heisenberg Uncertainty Principle
Appendix D EPR Thought Experiment
Appendix ? Photon Correlation Experiment
Appendix F Tables of Some 3-j Coefficients
Appendix G Proof of the Wigner-Eckart Theorem
Index