GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Volume 8

Section: Mathematics of Physics Peter A. Carruthers, Section Editor

Angular Momentum in Quantum Physics Theory and Application

L. C. Biedenharn Physics Department Duke University Durham, North Carolina

J. D. Louck Los Alamos National Laboratory University of California Los Alamos, New Mexico

With a Foreword by

Peter A. Carruthers Los Alamos National Laboratory University of California Los Alamos, New Mexico

CAMBRIDGE UNIVERSITY PRESS

Contents

Contents of companion volume The Racah-Wigner Algebra in Quantum Theory by L. C. Biedenharn and J. D. Louck (ENCYCLOPEDIA OF MATHEMATICS AND ITS

APPLICATIONS, Volume 9) xvii Editor's Statement xxiii Section Editor's Foreword xxv Preface xxvii Acknowledgments xxxi

PARTI

Chapter 1 Introduction 1 Notes 4 References 5

Chapter 2 The Kinematics of Rotations 7

1. Introduction 7 2. Properties of Rotations . . .7 3. Dirac's Construction 10 4. Cartan's Definition of a Spinor 15 5. Relation between SU(2) and SO{3) Rotations 17 6. Parametrizations of the Group of Rotations 19 7. Notes 25

References 26

Chapter 3 Standard Treatment of Angular Momentum in Quantum Mechanics 29

1. Overview 29 2. Defimtion of the Angular Momentum Operators 29 3. The Angular Momentum Multiplets 31 4. Matrices of the Angular Momentum 37 5. The Rotation Matrices (General Properties) 39 6. The Rotation Matrices (ExpUcit Forms) 46

xi

xii Contents

7. Wave Functions for Angular Momentum Systems 55 8. Differential Equations for the Rotation Matrices 57 9. Orthogonality of the Rotation Matrices 66

10. Spherical Harmonics 68 11. The Addition of Angular Momentum 71 12. The Wigner Coefficients 75 13. Relations between Rotation Matrices and Wigner

Coefficients 84 14. Concept of a Tensor Operator 87 15. The Wigner-Eckart Theorem 94 16. The Coupling of Tensor Operators 97 17. Applications of the Wigner-Eckart Theorem 99 18. Racah Coefficients 106 19. 9-y Coefficients 127 20. Rotationally Invariant Products 131 21. Operators Associated with Wigner, Racah, and

9-7 Coefficients 133 22. Notes 141 23. Appendices 162

References 175

Chapter 4 The Theory of Turns Adapted from Hamilton 180

1. An Alternative Approach to Rotations 180 2. Properties of Turns (Geometric Viewpoint) 184 3. Properties of Turns (Algebraic View) 193 4. The Space of Turns as a Carrier Space 198 5. Notes 201

References 203

Chapter 5 The Boson Calculus Applied to the Theory of Turns 205

1. Introduction 205 2. Excursus on the Boson Calculus 206 3. The Jordan Mapping 212 4. An Application of the Jordan Map 214 5. Generalization of the Jordan Map 217 6. Application of the Generalized Jordan Map 219 7. Application of the Generalized Jordan Map to

Determine the Wigner Coefficients 223 8. Wigner Coefficients as "Discretized" Rotation Matrices . . . 2 2 6 9. Appendices 230

References 264

о

Contents xiii

Chapter 6 Orbital Angular Momentum and Angular Functions on the Sphere 269

1. Rotational Symmetry of a Simple Physical System 270 2. Scalar Product of State Vectors 271 3. Unitarity of the Orbital Rotation Operator 272 4. A (Dense) Subspace of %(S) 273 5. Only Integral Values of / can occur in the

Quantization of Spatial (Orbital) Angular Momentum 274

6. Transformations of the Solid Harmonics under Orbital Rotation 275

7. The Elements of the Rotation Matrix ^'(R) are Homogeneous Polynomials 277

8. The Energy Eigenvalue Equation 278 9. Tensor Spherical Harmonics 279

10. Spinor Spherical Harmonics 283 11. Vector Spherical Harmonics 284 12. Algebraic Aspects of Vector Spherical Harmonics 286 13. Summary of Properties of Vector Solid Harmonics 292 14. Decomposition Theorem for Vector Functions Defined

on the Sphere 299 15. Rotationally Invariant Spherical Functions of Two

Vectors 302 16. Applications of the Cartan Map to Spherical Functions . . .305 17. Rotationally Invariant Spherical Functions in

Several Vectors 307 18. Relationship of Solid Harmonics to Potential Theory 311 19. The Orbital Rotation Matrices as Forms 313 20. The Orbital Rotation Matrices are Equivalent to Real

Orthogonal Matrices 316 21. The "Double-Valued Representations" of the Proper

Orthogonal Group SO(3) 316 22. Note 319

References 322

PART II

Chapter 7 Some Applications to Physical Problems 324

1. Introductory Remarks 324 2. Basic Principles Underlying the Applications 324 3. The Zeeman Effect 326

xiv Contents

a. Background 326 b. The Normal Zeeman Effect 327 с Quantal Treatment 327 d. The Anomalous Zeeman Effect 331 e. Relation to the Development of Angular

Momentum Theory 333 f. Concluding Remarks 334 g. Note 335 References 335

4. The Nonrelativistic Hydrogen Atom 335 a. Algebraic Aspects 336 b. Properties of the Bound States of the Hydrogen

Atom 340 с Explicit Hydrogen Atom Wave Functions 344 d. Momentum Space Representation 347 e. Relationship between Rotation Matrices and

Hyperspherical Harmonics 351 f. Pauli Particle (Hydrogen Atom with Spin) 353 g. Remarks 359 h. Appendix 361 References 362

5. Atomic Spectroscopy 364 a. Introduction 364 b. The Approximate Hamiltonian for Many-Electron

Atoms 365 с The Central-Field Model 366 d. A Short Vocabulary of Spectroscopy Terminology . . . . 376 e. Closed Shells 377 f. The One-Electron Problem with Spin-Orbit

Coupling 378 g. Two-Electron Configurations 381 h. Equivalent Electron Configurations 393 i. Operator Structures in /"-Configurations 406 j . Appendix 421 References 428

6. Electromagnetic Processes 432 a. Preliminary Remarks 432 b. Multipole Radiation 433 с The Hansen Multipole Fields 435 d. Classical Multipole Moments 437 e. Reduction of the Electric Multipole Moments 438 f. The Radiated Multipole Fields 439 g. A Curious Property of the Multipole Expansion

(Casimir [9]) 439

Contents xv

h. The Radiated Power 440 i. Angular Momentum Flux 441 j . A Vectorial Analog to the Rayleigh Expansion 442 k. An Illustrative Example 443 1. The Density Matrix for Photon Angular Correlation

Measurements 449 m. Notes 452 References 453

7. Angular Momentum Techniques in the Density Matrix Formulation of Quantum Mechanics 455 a. Preliminaries 455 b. Statistical Tensors 457 с A Geometric Characterization of the Density Matrices

for Pure States of Spin-y 463 d. The Density Matrix for a Relativistic Massive

Particle of Spin-y 467 e. The Special Case of Massless Particles 469 f. Coupling of Statistical Tensors 470 g. Some Examples Illustrating the Coupling Formula . . . . 471 h. The Majorana Formula 474 References 476

8. Angular Correlations and Angular Distributions of Reactions 478 a. The Nature of the Angular Correlation Process 478 b. Cascades 483 с Stretched Angular Momenta 484 d. More Involved Correlation Processes 484 e. Relativistic Regime 488 References 491

9. Some Applications to Nuclear Structure 492 a. Qualitative Considerations 492 b. The Nuclear Shell Model of Mayer and Jensen 493 с The Isospin Quantum Number 499 d. Properties of a Short-Range Interaction 500 e. The Pairing Interaction (Seniority) 508 f. Quasi-Spin 510 g. Quasi-Spin Wave Functions (Seniority Label) 512 h. Application of Quasi-Spin to Tensor Operators 515 i. Seniority in Terms of Casimir Operators 517 j . Concluding Remarks 523 k. Note 524 References 525

10. Body-Fixed Frames: Spectra of Spherical Top Molecules 527

xvi Contents

a. Introduction 527 b. Definition and Kinematics of a Body-Fixed Frame . . . . 528 с Form of the State Vectors for Isolated Systems

Described in a Body-Fixed Frame 533 d. The Instantaneous Principal Axes of Inertia Frame . . . . 534 e. The Eckart Molecular Frame 535 f. Distinguished Particle Frames 539 g. Uniform Method of Defining Body-Fixed Frames . . . . 540 h. Internal Coordinates 543 i. Internal Coordinates for the Eckart Frame 544 j . Internal Coordinates for the Principal Axes Frame . . . .549 k. The Linear Momentum Operators 554 1. The Hamiltonian for a Semirigid (Rigid) Polyatomic

Molecule 559 m. Approximate Form of the Hamiltonian for

Spherical Top Molecules 563 n. First-Order Energy Spectrum of a Triply Degenerate

Vibration in a Spherical Top Molecule 572 o. The Point Group of a Rigid Molecule 578 p. Higher-Order Corrections: Phenomenological

Hamiltonian 587 q. Splitting Patterns 594 r. Symmetry Axes and Induced Representations 602 s. High Angular Momentum Effects 609 t. Selection Rules and Statistical Weights 612 u. Spectra of Fundamental Transitions of SF6 624 v. Appendices 626 References 629

Appendix of Tables 634

Bibliography 667

List of Symbols 670

Author Index 687

Subject Index 695