Angular Momentum in Quantum Physics

8/19/2019 Angular Momentum in Quantum Physics

http://slidepdf.com/reader/full/angular-momentum-in-quantum-physics 1/7

GIAN-CARLO ROTA, EditorENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Volume 8

Section: Mathematics of PhysicsPeter A. C arruthers, Section Editor

Angular Momentumin Quantum PhysicsTheory and ApplicationL. C. BiedenharnPhysics DepartmentDuke University

Durham, N orth Carolina

J. D. LouckLos Alamos National LaboratoryUniversity of CaliforniaLos Alamos, New Mexico

With a Foreword by

Peter A. CarruthersLos Alamos National LaboratoryUniversity of CaliforniaLos Alamos, New Mexico

C A M B R I D G EU N I V E R S I T Y P R E S S



ontents

Contents of companion volumeThe Racah-Wigner Algebra in Quantum Theoryby L. C. Biedenharn and J. D. Louck(ENCYCLOPEDIA OF MATHEMATICS AND ITS

APPLICATIONS, Volume 9) xviiEditor s Statement xxiiiSection Editor s Foreword xxvPreface xxviiAcknowledgments xxxi

PARTI

Chapter Introduction 1Notes 4References 5

Chapter 2 The Kinematics of Rotations 7

1. Introduction 72. Properties of Rotations . . .73. Dirac 's Construction 104. Cartan 's Definition of a Spinor 155. Relation between SU(2) and SO{3 Rotations 176. Parametrizations of the Group of Rotations 197. Notes 25

References 26

Chapter 3 Standard Treatment of Angular Momentum inQuantum Mechanics 29

1. Overview 292. Defimtion of the Angular Momentum Operators 293. The Angular Momentum Multiplets 314. Matrices of the Angular Momentum 375. The Rotation Matrices (General Properties) 396. The Rotation Matrices (ExpUcit Forms) 46

x



xii Contents

7. Wave Fu nction s for An gular M om entum Systems 558. Differential Eq ua tions for the Rota tion M atrices 579. Orthog onality of the Ro tation M atrices 66

10. Spherical Ha rm onics 6811. The Add ition of Angular M om entum 7112. Th e W igner Coefficients 7513. Relations between R otation Matrices and W igner

Coefficients 8414. Con cept of a Tenso r Op erator 8715. The W igner-Eckart Theorem 9416. The Coup ling of Tensor Op erators 9717. Ap plications of the W igner-Eckart Theo rem 9918. Ra cah Coefficients 10619. 9-y Coefficien ts 12720. Ro tationally Invarian t Produ cts 13121. Op erators Associated with Wigner, Racah, and

9-7 Coefficients 13322. N otes 14123. Ap pendices 162

References 175

Chapter Th e Theory of Turns Adapted from Ham ilton 180

1. An Alternative Ap proach to Ro tations 1802. Properties of Tu rns (Geo m etric View point) 1843. Properties of Tu rns (Algebraic View) 1934. Th e Space of Tu rns as a Carrier Space 1985. No tes 201

References 203

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

1. Introd uctio n 2052. Excursus on the Boson Calculus 2063. The Jordan M apping 2124. An Ap plication of the Jordan M ap 2145. Ge neralization of the Jord an M ap 2176. Ap plication of the Gene ralized Jor da n M ap 2197. Ap plication of the Gen eralized Jor dan M ap to

D eterm ine the W igner Coefficients 2238. Wigner Coefficients as Discretized Rotation Matrices . . . 2 2 69. Ap pendices 230

References 264



Contents xiii

Chapter 6 Orbital Angular Momentum and Angular Functionson the Sphere 269

1. R otationa l Sym metry of a Simple Physical System 270

2. Scalar Product of State Vec tors 2713. U nitarity of the Orb ital Ro tation O pera tor 2724. A (Dense) Subspace of (S ) 2735. Only Integra l Values of / can occur in the

Quantization of Spatial (Orbital)Angular Mom entum 274

6. Tran sform ations of the Solid Ha rm onics und erOrb ital Ro tation 275

7. The Elem ents of the Ro tation M atrix ^'(R) are

Hom ogeneous Polynomials 2778. Th e Energy Eigenvalue Equ ation 2789. Ten sor Spherical H arm onics 279

10. Spinor Spherical Ha rm onics 28311. Vector Spherical H arm onics 28412. Algebraic Aspec ts of Vector Spherical Ha rm onics 28613. Sum m ary of Prope rties of Vector Solid Ha rm onics 29214. Decom position Theorem for Vector Fun ctions Defined

on the Sphere 299

15. Ro tationally Invarian t Spherical Fu nctio ns of Tw oVectors 302

16. Ap plications of the C artan Ma p to Spherical Fu nctio ns . . .30517. Ro tationally Invariant Spherical Fu nctio ns in

Several Vec tors 30718. Re lationship of Solid Ha rm onics to Potential Theo ry 31119. The Orbital Ro tation Matrices as Form s 31320. Th e Orb ital Ro tation Matrices are Equ ivalent to Real

Ortho gon al M atrices 316

21. The Do uble-Valued Re presen tations of the ProperOrthogonal Group SO(3) 316

22. N ote 319References 322

PART II

Chapter 7 Som e Applications to Physical Problems 324

1. Introductory Rem arks 3242. Basic Principles Un derlying the Ap plications 3243. Th e Zeem an Effect 326



xiv Contents

a. Background 326b. The Norm al Zeeman Effect 327с Quantal Treatment 327d. The Anomalous Zeeman Effect 331

e. Relation to the Development of AngularMomentum Theory 333f. Concluding Remarks 334g. Note 335References 335

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

Atom 340

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

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

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

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

Coupling 378g. Two-Electron Configurations 381h. Equivalent Electron Configurations 393i. Operator Structures in / -Configurations 406

Appendix 421References 428

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

(Casimir [9] 439



Contents xv

h. Th e Rad iated Power 440i. Angular M om entum Flux 441

A Vectorial An alog to the Rayleigh Expan sion 442k. An Illustrative Exam ple 443

1. Th e Den sity M atrix for Pho ton An gular CorrelationMeasurements 449m. No tes 452References 453

7. Ang ular M om entum Techniques in the Density M atrixForm ulation of Qu antum Mechanics 455a. Prelim inaries 455b. Statistical Te nso rs 457с A Geo m etric Characterization of the Density Matrices

for Pu re Sta tes of Spin-y 463d. The Density M atrix for a Relativistic M assiveParticle of Spin-y 467

e. Th e Special Case of Massless Particles 469f. Co uplin g of Statistical Ten sors 470g. Some Examples Illustrating the Coupling Form ula . . . . 471h. The Majorana Form ula 474References 476

8. An gular Co rrelations and An gular D istributions of

Reactions 478a. Th e N atu re of the An gular Co rrelation Process 478b. Cascad es 483с Stretched Angular M om enta 484d. M ore Involved Co rrelation Processes 484e. Rela tivistic Re gime 488References 491

9. Some A pplications to Nu clear Structure 492a. Qu alitative Co nsiderations 492

b . Th e Nu clear Shell Mo del of M ayer and Jensen 493с The Isospin Qu antum Nu mb er 499d. Prop erties of a Sho rt-Rang e Interaction 500e. Th e Pairing Inter action (Seniority) 508f. Qu asi-Spin 510g. Quasi-Spin W ave Fu nction s (Seniority Label) 512h. A pplication of Quasi-Spin to Tenso r O perato rs 515i. Seniority in Term s of Casim ir O perato rs 517

Concluding Rem arks 523

k. N ote 524References 525

10. Body-Fixed Fram es: Spectra of Spherical To pMolecules 527



xvi Contents

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

Described in a Body-Fixed Frame 533

d. The Instantaneous Principal Axes of Inertia Frame . . . . 534e. The Eckart Molecular Frame 535f. Distinguished Particle Frames 539g. Uniform Method of Defining Body-Fixed Frames . . . . 540h. Internal Coordinates 543i. Internal Coordinates for the Eckart Fram e 544

Internal Coordinates for the Principal Axes Frame . . . .549k. The Linear Momentum Operators 5541. The Hamiltonian for a Semirigid (Rigid) Polyatomic

Molecule 559m. Approximate Form of the Hamiltonian for

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

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

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

Appendix of Tables 634

Bibliography 667

List of Symbols 670

Author Index 687

Subject Index 695

Documents

Angular Momentum in Quantum Physics