Group Theory in Physics (eBook)

An Introduction

John F. Cornwell (Autor)

eBook Download: PDF

1997 | 1. Auflage
349 Seiten
Elsevier Science (Verlag)
978-0-08-053266-0 (ISBN)

Infos

This book, an abridgment of Volumes I and II of the highly respected Group Theory in Physics, presents a carefully constructed introduction to group theory and its applications in physics. The book provides anintroduction to and description of the most important basic ideas and the role that they play in physical problems. The clearly written text contains many pertinent examples that illustrate the topics, even for those with no background in group theory.
This work presents important mathematical developments to theoretical physicists in a form that is easy to comprehend and appreciate. Finite groups, Lie groups, Lie algebras, semi-simple Lie algebras, crystallographic point groups and crystallographic space groups, electronic energy bands in solids, atomic physics, symmetry schemes for fundamental particles, and quantum mechanics are all covered in this compact new edition.

Key Features
* Covers both group theory and the theory of Lie algebras
* Includes studies of solid state physics, atomic physics, and fundamental particle physics
* Contains a comprehensive index
* Provides extensive examples

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This book, an abridgment of Volumes I and II of the highly respected Group Theory in Physics, presents a carefully constructed introduction to group theory and its applications in physics. The book provides anintroduction to and description of the most important basic ideas and the role that they play in physical problems. The clearly written text contains many pertinent examples that illustrate the topics, even for those with no background in group theory.This work presents important mathematical developments to theoretical physicists in a form that is easy to comprehend and appreciate. Finite groups, Lie groups, Lie algebras, semi-simple Lie algebras, crystallographic point groups and crystallographic space groups, electronic energy bands in solids, atomic physics, symmetry schemes for fundamental particles, and quantum mechanics are all covered in this compact new edition. - Covers both group theory and the theory of Lie algebras- Includes studies of solid state physics, atomic physics, and fundamental particle physics- Contains a comprehensive index- Provides extensive examples

Inhalt

Front Cover 1
Group Theory in Physics: An Introduction 2
Copyright Page 3
Contents 4
Preface 8
Chapter 1. The Basic Framework 12
1. The concept of a group 12
2. Groups of coordinate transformations 15
3. The group of the Schrödinger equation 21
4. The role of matrix representations 26
Chapter 2. The Structure of Groups 30
1. Some elementary considerations 30
2. Classes 32
3. Invariant subgroups 34
4. Cosets 35
5. Factor groups 37
6. Homomorphic and isomorphic mappings 39
7. Direct products and semi-direct products of groups 42
Chapter 3. Lie Groups 46
1. Definition of a linear Lie group 46
2. The connected components of a linear Lie group 51
3. The concept of compactness for linear Lie 53
4. Invariant integration 55
Chapter 4. Representations of Groups - Principal Ideas 58
1. Definitions 58
2. Equivalent representations 60
3. Unitary representations 63
4. Reducible and irreducible representations 65
5. Schur's Lemmas and the orthogonality theorem for matrix representations 68
6. Characters 70
Chapter 5. Representations of Groups - Developments 76
1. Projection operators 76
2. Direct product representations 81
3. The Wigner-Eckart Theorem for groups of coordinate transfor-mations in IR3 84
4. The Wigner-Eckart Theorem generalized 90
5. Representations of direct product groups 94
6. Irreducible representations of finite Abelian groups 96
7. Induced representations 97
Chapter 6. Group Theory in Quantum Mechanical Calculations 104
1. The solution of the Schrödinger equation 104
2. Transition probabilities and selection rules 108
3. Time-independent perturbation theory 111
Chapter 7. Crystallographic Space Groups 114
1. The Bravais lattices 114
2. The cyclic boundary conditions 118
3. Irreducible representations of the group T of pure primitive translations and Bloch's Theorem 120
4. Brillouin zones 122
5. Electronic energy bands 126
6. Survey of the crystallographic space groups 129
7. Irreducible representations of symmorphic space groups 132
8. Consequences of the fundamental theorems 140
Chapter 8. The Role of Lie Algebras 146
1. "Local" and "global" aspects of Lie groups 146
2. The matrix exponential function 147
3. One-parameter subgroups 150
4. Lie algebras 151
5. The real Lie algebras that correspond to general linear Lie groups 156
Chapter 9. The Relationships between Lie Groups and Lie Algebras Explored 164
1. Introduction 164
2. Subalgebras of Lie algebras 164
3. Homomorphic and isomorphic mappings of Lie algebras 165
4. Representations of Lie algebras 171
5. The adjoint representations of Lie algebras and linear Lie groups 179
6. Direct sum of Lie algebras 182
Chapter 10. The Three-dimensional Rotation Groups 186
1. Some properties reviewed 186
2. The class structures of SU(2) and SO(3) 187
3. Irreducible representations of the Lie algebras su(2) and so(3) 188
4. Representations of the Lie groups SU(2), SO(3) and O(3) 194
5. Direct products of irreducible representations and the Clebsch-Gordan coefficients 197
6. Applications to atomic physics 200
Chapter 11. The Structure of Semi-simple Lie Algebras 204
1. An outline of the presentation 204
2. The Killing form and Cartan's criterion 204
3. Complexification 209
4. The Cartan subalgebras and roots of semi-simple complex Lie algebras 211
5. Properties of roots of semi-simple complex Lie algebras 218
6. The remaining commutation relations 224
7. The simple roots 229
8. The Weyl canonical form of L 234
9. The Weyl group of L 235
10. Semi-simple real Lie algebras 239
Chapter 12. Representations of Semi-simple Lie Algebras 246
1. Some basic ideas 246
2. The weights of a representation 247
3. The highest weight of a representation 252
4. The irreducible representations of L = A2, the complexification of L = su(3) 256
5. Casimir operators 262
Chapter 13. Symmetry schemes for the elementary particles 266
1. Leptons and hadrons 266
2. The global internal symmetry group SU(2) and isotopic spin 267
3. The global internal symmetry group SU(3) and strangeness 270
APPENDICES 280
Appendix A. Matrices 282
1. Definitions 282
2. Eigenvalues and eigenvectors 286
Appendix B. Vector Spaces 290
1. The concept of a vector space 290
2. Inner product spaces 293
3. Hilbert spaces 297
4. Linear operators 299
5. Bilinear forms 303
6. Linear functionals 305
7. Direct product spaces 306
Appendix C. Character Tables for the Crystallographic Point Groups 310
D. Properties of the Classical Simple Complex Lie Algebras 330
1. The simple complex lie algebra Al, l = 1 330
2. The simple complex Lie algebra Bl, l = 1 331
3. The simple complex Lie algebra Cl, 1 = 1 333
4. The simple complex Lie algebra D1, 1 = 3 (and the semi-simple complex Lie algebra D2) 335
References 338
Index 346

Produkt-Details

• Erscheint lt. Verlag: 11.7.1997
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 0-08-053266-7 / 0080532667
• ISBN-13: 978-0-08-053266-0 / 9780080532660