An Introduction to Groups and their Matrices for Science Students (Instructor Solution Manual, Solutions)

GROUPS
	Contents
	PREFACE
	INTRODUCTION
	OVERVIEW
	TO THE INSTRUCTOR
	FUNDAMENTAL CONCEPTS
	1.1	Introduction
	1.2	Operations
	1.2.1 Symmetry Operations
	1.2.2 Products of Operations
	1.2.3 Product Tables
	1.2.4 The Inverse of an Operation
	EA
	A A A2
	EA D E
	ADE
	EE
	EA
	E E A
	AAE
	1.3 What Is a Group?
	1.3.1 Discrete and Continuous Groups
	1.4	Examples of Groups
	1.4.1 Abelian Groups
	1.4.2 The 32 Group
	1.4.3 The Permutation (Symmetric) Group
	Products of Permutations
	Cycle Notation
	1.5	Matrix Representations of Groups
	1.5.1 Isomorphism
	CDF
	1.5.2 Homomorphism
	EAB
	E A B CDF
	1.5.3 An Example of a Matrix Representation
	1.6 Matrix Algebra
	1.6.1 A Constant Times a Matrix
	1.6.2 Addition of Matrices
	1.6.3 Products of Matrices
	1.6.4 Determinant of a Matrix
	1.6.5 The Kronecker Delta
	1.7	Special Matrices
	1.7.1 The Complex Conjugate of a Matrix
	1.7.2 The Transpose of a Matrix
	1.7.3 The Adjoint of a Matrix
	1.7.4 Hermitian Matrices
	1.7.5 Unitary and Orthogonal Matrices
	1.8	A Brief History of Group Theory
	1.9	Brief Bios
	Summary of Chapter 1
	Problems and Exercises
	MATRIX
	REPRESENTATIONS OF
	DISCRETE GROUPS
	2.1	Introduction
	2.2	Basis Functions and Representations
	ET
	TTE
	2.2.1 “Transforms Like . . . ”
	2.3	Similarity Transformations
	2.4	Equivalent Representations
	2.5 Similarity Transformations and Unitary Matrices
	2.6 Character and Its Invariance under Similarity Transformations
	ET
	2.7 Irreducible Representations
	2.7.1 The Regular Representation
	2.7.2 Example: Reducing the Regular Representation
	EABC DF
	EA B C D F
	CD
	2.7.3 Conjugates and Classes
	2.7.4 Orthogonality Theorems
	Theorem 5
	Theorem 6
	Theorem 7	2
	Theorem 8
	Tests for Reducibility
	An Easy Way to Find Character Tables
	2.8 Kronecker (Direct) Product
	2.8.1 Example: The Klein Four-Group
	2.9 Kronecker Sum
	Summary of Chapter 2
	Stated Theorems in Chapter 2
	Problems and Exercises
	SD
	MOLECULAR VIBRATIONS
	3.1 Introduction
	3.2 Oscillating Systems and Newton’s Laws
	3.2.1 Normal Modes
	3.2.2 Example: Two Masses and Three Springs
	3.3 Normal Modes and Group Theory
	3.3.1 Degeneracy
	3.4	Normal Modes of a Water Molecule
	3.5	Visualizing Normal Modes
	3.5.1 Projection Operator
	An Easy Way to Find Basis Functions
	3.5.2 Visualizing the Water Molecule’s Normal Modes
	3.6 Infrared (IR) Spectroscopy
	3.6.1 Thermal Radiation
	3.7 Raman Spectroscopy
	3.8 Brief Bios
	Summary of Chapter 3
	Problems and Exercises
	CRYSTALLINE SOLIDS
	4.1 Introduction
	4.2 Bravais Lattices
	4.2.1 Translation Symmetry and Basis Vectors
	4.3 X-Ray Crystallography
	4.4 Fourier Transform
	4.5 Reciprocal Lattice
	4.5.1 Miller Indices
	Powder XRD
	4.6 Lattice Translation Group
	4.6.1 Bloch’s Theorem
	4.6.2 Quantum Mechanics of Crystals
	4.7 Crystallographic Point Groups and Rotation
	Symmetry
	4.7.1 Plane Diagrams of Group Symmetry Operations
	4.8 Crystallographic Space Groups and the Seitz Operator
	4.9 Crystal Symmetry Operations
	4.9.1 Notation
	4.9.2 Cubic Lattice Character Table
	4.10	Lattice Vibrations
	4.10.1 Diatomic Linear Chain and Dispersion
	4.10.2 Visualizing Lattice Vibrations
	4.10.3 Phonons
	4.11	Brief Bios
	Summary of Chapter 4
	Problems and Exercises
	BOHR’S QUANTUM THEORY AND MATRIX MECHANICS
	5.1	Introduction
	5.2	Bohr’s Model
	5.2.1 Bohr’s Correspondence Principle
	5.3	Matrix Mechanics
	5.3.1 Some Concepts
	The Hamiltonian
	Fourier Series
	5.3.2 The Beginnings of Matrix Mechanics - From Continuous to Discrete
	5.4	Matrix Mechanics Quantization
	5.4.1 Matrix Mechanics Dynamics
	5.4.2 Quantization Condition
	5.4.3 Time and the Hamiltonian
	5.5	Consequences of Matrix Mechanics
	5.5.1 Conservation of Energy
	5.5.2 Bohr Frequency Condition
	5.5.3 Conservation of Angular Momentum
	MDrp
	5.6 Heisenberg Uncertainty Relation
	5.7 Brief Bios
	Summary of Chapter 5
	Problems and Exercises
	WAVE MECHANICS, MEASUREMENT, AND ENTANGLEMENT
	6.1	Introduction
	6.2	Schrodinger’s Wave Mechanics
	6.2.1 Wave Packets
	6.2.2 Two-Slit Interference
	6.3	The Wave Equation
	6.3.1 Eigenvalues, Eigenvectors, and Matrix Elements
	6.3.2 Dirac’s Bra-Ket Bracket Notation
	6.4 Quantization Conditions in Wave Mechanics
	6.4.1 Example: Rigid Rotor (Dumbbell)
	6.4.2 Example: Rigid Rotor (Diatomic Molecule)
	6.4.3 Rotational Spectra of Diatomic Molecules
	6.5 Matrix Diagonalization
	6.5.1 Example: Diagonalizing a Matrix
	6.6	Quantum Measurement
	6.6.1 Born’s Rule
	6.7	The EPR Paradox and Entanglement
	6.7.1 Entangled Wave Functions
	6.7.2 Hidden Variables
	6.7.3 Correlation of Entangled Photons: Hidden Variables
	6.7.4 Correlation of Entangled Photons: Quantum Mechanics
	6.8 Brief Bio
	Summary of Chapter 6
	Problems and Exercises
	ROTATION
	7.1	Introduction
	7.2	Two Ways of Looking at Rotation
	7.2.1 Rotated Vector, Fixed Axes
	Rotation Notation
	7.2.2 Rotated Axes, Fixed Vector
	7.2.3 Inverse of a Rotation
	7.2.4 Rotation about an Arbitrary Axis
	7.3	Rotation of a Function
	7.4	The Axial Rotation Group
	7.4.1 Vibration of a Hydrogen Molecule
	7.5 1he U(1) and SU(2) Groups
	7.5.1 U(1)
	7.5.2 SU(2)
	The Importance of U(1) and SU(2)
	7.6 Pauli Matrices, SU(2), and Rotation
	°z uC	V0 - V W D 2\\°Л
	7.6.1 Spinors
	7.7	Euler Angles
	7.7.1 Euler Angles and Stationary Axes
	7.7.2 Euler Angles and Rotated Axes
	7.7.3 Problems with Euler Angle Applications
	0A
	0A
	7.8	Finite Rotations Don’t Commute
	7.9	. . . But Rotations Do Commute to First Order
	7.10 Brief Bios
	Summary of Chapter 7
	Problems and Exercises
	QUANTUM ANGULAR MOMENTUM
	8.1	Introduction
	8.2	Stern and Gerlach: An Important Experiment
	(1922)
	8.2.1 The Apparatus
	8.2.2 Magnetic Moment in a Magnetic Field
	8.2.3 The Result
	8.2.4 Quantum Numbers
	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
	8.7.1 Matrix Representation of D .i/
	2( i
	8.7.2 Rotation Matrices and SU(2)
	8.8 The ujm Are Spherical Harmonics
	8.8.1 Spherical Harmonics and Group Theory
	Y1 .
	8.8.2 Spherical Harmonics and Differential Equations
	8.9	Spin Basis Functions and Pauli Matrices
	8.10	Coupling (Adding) Angular Momenta
	8.10.1 Example: Positronium
	8.10.2 Wigner 3-j Coefficients
	8.10.3 Clebsch-Gordan Coefficients
	8.11	Wigner-Eckart Theorem
	8.11.1 Theorem and Notation
	8.11.2 How It Is Used
	8.12	Selection Rules
	8.12.1 From Spherical Harmonics
	8.12.2 From the Wigner-Eckart Theorem
	8.12.3 From Parity-Inversion
	8.13 Brief Bios
	Summary of Chapter 8
	Problems and Exercises
	THE STRUCTURE OF
	ATOMS
	9.1	Introduction
	9.2	Zeeman: An Important Experiment (1897)
	9.2.1 Classical Theory of the Zeeman Effect
	Magnetic Fields of Sunspots
	9.3	Quantum Theory of the Zeeman Effect
	9.3.1 Weak Field Zeeman
	S D J - L
	L D J - S
	9.3.2 Term Notation and Zeeman Splitting
	9.3.3 Strong Field (Paschen-Back) Zeeman
	9.4	Fine Structure
	S D J - L
	9.4.1 Making Estimates
		9.5	Example: Intermediate Field Zeeman
	9.6 Nuclear Spin and Hyperfine Structure
	FDICJ
	9.6.1 Hyperfine Splitting
	1 H and Radio Astronomy
	9.6.2 Hyperfine Structure Zeeman
	Weak Field
	\' J J - F	I I - F \\
	F=JCI
	9.7	Multi-electron Atoms
	9.7.1 The Pauli Exclusion Principle
	9.8	The Helium Atom
	9.8.1 Perturbation Theory
	9.9 The Structure of Multi-electron	parahf,um
	Atoms
	9.9.1 Electron Configurations and the States of Atoms
	P4 P5 Рб
	Slater Determinant Wave Functions
	9.9.2 Electron Configuration and the Building-Up Principle
	Summary of Chapter 9
	Problems and Exercises
	10
	PARTICLE PHYSICS
	10.1	Introduction
	10.1.1 The Fundamental Forces
	10.2	Natural Units
	10.2.1 Converting between Systems of Units
	10.3 Isospin
	10.3.1 The Nucleon Isospin Doublet
	Binding Energy and Mirror Nuclei
	10.3.2 The Deuteron
	10.3.3 The Pion Isospin Triplet
	10.3.4 The Sigma Isospin Triplet
	10.3.5 The Delta Isospin Quadruplet
	10.4 Cross Section
	10.4.1 Pion-Nucleon Scattering
	10.5	Antiparticles
	10.6	The Lagrangian
	10.6.1 Stationary Action
	10.6.2 The Lagrangian and Invariance (Noether’s Theorem)
	10.7	Gauge Theory
	10.7.1 Local Gauge Invariance of Schrodinger’s Equation
	10.7.2 Lagrangian, Gauge Theory, and Particle Physics
	10.8 All Those Particles - the Particle Zoo
	10.9 The Quark Model
	10.9.1 Classes of Elementary Particles
	10.9.2 The Charm Quark
	10.9.3 Color Charge
	10.10 Conservation Laws and Quantum Numbers
	10.10.1 Examples: Particle Reactions
	10.11 Group Theory and Particle Physics
	10.11.1 SU(3)
	10.11.2 Octet and Decuplet Particle Multiplets
	333 D 10881:
	10.11.3 SU(3) and Color Charge
	10.12 Concluding Remark
	Summary of Chapter 10
	Problems and Exercises
	Appendix A
	Character Tables from Class
	Sums
	Appendix B
	Born-Jordan Proof of the
	Quantization Condition
	Appendix C
	Weyl Derivation of the Heisenberg Uncertainty Principle
	Appendix D
	EPR Thought Experiment
	Appendix E
	Photon Correlation
	Experiment
	Appendix F
	Tables of Some 3-j Coefficients
	Appendix G
	Proof of the Wigner-Eckart Theorem
	Index