Symmetries in Quantum Physics
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Table of contents
- Contentsv
- Prefacexiii
- Chapter 1. Introduction1
- Transformation Theories: Klein's and Dirac's3
- 1.1 Symmetry and the Selection of Variables5
- 1.2 Algebraic Elements11
- 1.3 Reduction Procedure and Irreducible Tensorial Sets17
- 1.4 Further Aspects of Reduction20
- 1.5 Structure of the Book26
- 1.6 Quaternions28
- Problems29
- PART A : STATE REPRESENTATIVES AND r-TRANSFORMATIONS: THEIR CONSTRUCTION AND PROPERTIES31
- Chapter 2. Infinitesimal Rotations and Angular Momentum33
- 2.1 Basic Relations34
- 2.2 Analytical Example: Infinitesimal Transformation of Cartesian Coordinates38
- 2.3 The Angular Momentum Matrices of Quantum Mechanics41
- 2.4 The Fundamental Representation45
- Problems49
- Chapter 3. Frame Reversal and Complex Conjugation51
- 3.1 Analytical Representation and Implications of Frame Reversal54
- 3.2 Contragredience and the Construction of Invariants62
- 3.3 Cartesian Base for Integer j >= 168
- Problems74
- Chapter 4. Standard r-Transformation Matrices and Their Applications75
- 4.1 Explicit Form and Properties76
- 4.2 Macroscopic Applications85
- 4.3 Applications to Quantum Physics88
- 4.4 Coordinate Inversion and Parity Eigenfunctions93
- Problems97
- Chapter 5. Reduction of Direct Products (Addition of Angular Momenta)99
- 5.1 Structure and Properties of the Reducing Matrix100
- 5.2 Reduction of r-Transformation Products110
- 5.3 Irreducible Product Sets112
- 5.4 Symmetrization of Wigner Coefficients:119
- Problems123
- PART B: TENSORIAL ASPECTS OF QUANTUM PHYSICS125
- Chapter 6. Tensorial Sets of Quantum Operators127
- 6.1 The Liouville Representation of Quantum Mechanics128
- 6.2 Quantum Mechanics of Particles with Spin 1/2130
- 6.3 Two-Level Systems135
- 6.4 Particles with Spin j > 1/2: Wigner-Eckart Theorem140
- 6.5 Systems with 2j + 1 Levels146
- 6.6 Transfer of Angular Momentum148
- 6.7 Calculation of Matrix Elements150
- Problems154
- Chapter 7. Recoupling Transformations: 6-j and 9-j Coefficients157
- 7.1 Transformation Matrices and Their Analysis160
- 7.2 Symmetrized Recoupling: 6-j and 9-j Coefficients168
- 7.3 Products of Operators176
- 7.4 Combining Operators of Different Systems183
- 7.5 Illustrations186
- Problems195
- Chapter 8. Partially Filled Shells of Atoms or Nuclei199
- 8.1 Qualitative Discussion200
- 8.2 Shell-wide Treatment208
- 8.3 Algebra of Triple Tensors and Its Applications214
- PART C: SYMMETRIES OF HIGHER DIMENSIONS225
- Chapter 9. Discrete Transformations of Coordinates229
- 9.1 Point Symmetry Operations and Their Groups230
- 9.2 Characters of Group Representations and Their Applications235
- 9.3 Symmetries of Molecules and Crystals243
- Problems250
- Chapter 10. Rotation Groups in Higher Dimensions: Multiparticle Problems251
- 10.1 Four-Dimensional Rotations: The Coulomb-Kepler Problem252
- 10.2 Orthogonal Groups in Higher Dimensions263
- 10.3 Further Developments272
- Chapter 11. Lorentz Transformations and the Lorentz and Poincaré Groups279
- 11.1 Lorentz Transformations281
- 11.2 Generators and Representations of the Lorentz Group283
- 11.3 The Inhomogeneous Lorentz (Poincaré) Group295
- 11.4 Field Representations297
- Chapter 12. Symmetries of the Scattering Continuum301
- 12.1 Symmetries of Radial Eigenfunctions302
- 12.2 The Full Noninvariance Group of Hydrogen305
- 12.3 Dynamics and Symmetry Transformations310
- Bibliography313
- Index317
This text focuses on the physics of symmetries, developing symmetries and transformations through concrete physical examples and contexts rather than presenting the information axiomatically, mathematically, and abstractly. Readers are introduced gradually to advanced mathematical procedures, including the Wigner and Racah algebras and their applications to various symmetry groups. The book also includes some of the latest research on the use of non-invariance and non-compact groups in the consideration of relativistic and many-particle problems of atoms and nuclei.
This book is an updated replacement for the text Irreducible Tensorial Sets (Academic Press, 1959). Parts A and B of the present book grew out of occasional lectures in the intervening decades at the University of Chicago, where it became neccessary to update or elaborate upon certain points. Part C has been built more recently to deal with innovations and new information in the field of mathematical physics. The book as a whole develops the subject of symmetry from a physical point of view, allowing students and researchers to gain new insight on their subject. This book can be used both as a text and as a reference by students and scientists in the field.
Adapts and extends the earlier Irreducible Tensor Sets (Academic Press, 1959) to classroom use
Extends to multi-particle systems and relativity
Includes problems in each chapter for homework assignments
Embraces the latest research on non-invariance groups
This book is an updated replacement for the text Irreducible Tensorial Sets (Academic Press, 1959). Parts A and B of the present book grew out of occasional lectures in the intervening decades at the University of Chicago, where it became neccessary to update or elaborate upon certain points. Part C has been built more recently to deal with innovations and new information in the field of mathematical physics. The book as a whole develops the subject of symmetry from a physical point of view, allowing students and researchers to gain new insight on their subject. This book can be used both as a text and as a reference by students and scientists in the field.
Adapts and extends the earlier Irreducible Tensor Sets (Academic Press, 1959) to classroom use
Extends to multi-particle systems and relativity
Includes problems in each chapter for homework assignments
Embraces the latest research on non-invariance groups