W. Greiner · B. Muller QUANTUM MECHANICS Symmetries

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With a Foreword by D. A. Bromley

Second Revised Edition With 81 Figures, and 128 Worked Examples and Problems

Springer Professor Dr. Walter Greiner Professor Dr. Berndt Muller Institut fiir Theoretische Physik der Physics Department Johann Wolfgang Goethe-Universitat Frankfurt Duke University Postfach I I 1932 P.O. Box 90305 60054 Frankfurt am Main Durham, NC 27708-0305 Germany USA Street address: email: [email protected] Robert-Mayer-Strasse 8-10 60325 Frankfurt am Main Germany email: [email protected]

Title of the original German edition: Theoretische Physik, Band 5: Quantenmechanik II, Symmetrien 3. Aujl. © Verlag Harri Deutsch, Thun, I 984, I 992

Second Edition I 994 Second Printing I 997 Third Printing 200 I

ISBN 3-540-58080-8 2. Auflage Springer-Verlag Berlin Heidelberg New York ISBN 3-540-19201-8 I. Autlage Springer-Verlag Berlin Heidelberg New York

This volume originally appeared in the series "Theoretical Physics -Text and Exercise Books Vol. 2".

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Typesetting: Data conversion by LE-TEX, Leipzig Cover design: Design Concept, Emil Smejkal, Heidelberg Copy Editor: V. Wicks, Heidelberg Production Editor: P. Treiber, Heidelberg Printed on acid-free paper SPIN 10786187 56/3141/tr- 54 3 2 I 0 Foreword to Earlier Series Editions

More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modem theoretical physics - with mathematics, the most fundamental of sciences -using Walter Greiner's textbooks as their guide. The idea of developing a coherent, complete presentation of an entire field of science in a series of closely related textbooks is not a new one. Many older physicists remember with real pleasure their sense of adventure and discovery as they worked their ways through the classic series by Sommerfeld, by Planck and by Landau and Lifshitz. From the students' viewpoint, there are a great many obvious advantages to be gained through use of consistent notation, logi• cal ordering of topics and coherence of presentation: beyond this, the complete coverage of the science provides a unique opportunity for the author to convey his personal enthusiasm and love for his subject. The present five volume set, Theoretical Physics, is in fact only that part of the complete set of textbooks developed by Greiner and his students that presents the quantum theory. I have long urged him to make the remaining volumes on classical mechanics and dynamics, on electromagnetism, on nuclear and particle physics, and on special topics available to an English-speaking audience as well, and we can hope for these companion volumes covering all of theoretical physics some time in the future. What makes Greiner's volumes of particular value to the student and profes• sor alike is their completeness. Greiner avoids the all too common "it follows that . . . " which conceals several pages of mathematical manipulation and confounds the student. He does not hesitate to include experimental data to illu• minate or illustrate a theoretical point and these data, like the theoretical content, have been kept up to date and topical through frequent revision and expansion of the lecture notes upon which these volumes are based. Moreover, Greiner greatly increases the value of his presentation by includ• ing something like one hundred completely worked examples in each volume. Nothing is of greater importance to the student than seeing, in detail, how the theoretical concepts and tools under study are applied to actual problems of inter• est to a working physicist. And, finally, Greiner adds brief biographical sketches to each chapter covering the people responsible for the development of the the• oretical ideas and/ or the experimental data presented. It was Auguste Comte ( 1798-1857) in his Positive Philosophy who noted, "To understand a science it is necessary to know its history". This is all too often forgotten in modern VI Foreword to Earlier Series Editions

physics teaching and the bridges that Greiner builds to the pioneering figures of our science upon whose work we build are welcome ones. Greiner's lectures, which underlie these volumes, are internationally noted for their clarity, their completeness and for the effort that he has devoted to mak• ing physics an integral whole; his enthusiasm for his science is contagious and shines through almost every page. These volumes represent only a part of a unique and Herculean effort to make all of theoretical physics accessible to the interested student. Beyond that, they are of enormous value to the professional physicist and to all others working with quantum phenomena. Again and again the reader will find that, after dipping into a particular volume to review a specific topic, he will end up browsing, caught up by often fascinating new insights and developments with which he had not previously been familiar. Having used a number of Greiner's volumes in their original German in my teaching and research at Yale, I welcome these new and revised English trans• lations and would recommend them enthusiastically to anyone searching for a coherent overview of physics.

Yale University D. Allan Bromley New Haven, CT, USA Henry Ford II Professor of Physics 1989 Preface to the Second Edition

We are pleased to note that our text Quantum Mechanics - Symmetries has found many friends among physics students and researchers so that the need for a second edition has arisen. We have taken this opportunity to make sev• eral amendments and improvements to the text. We have corrected a number of misprints and minor errors and have added explanatory remarks at various places. In addition to many other smaller changes the sections 8.6, 8.11, and 11.4 and the exercises 3.9, 7.8, and 9.5 have been expanded. Two new exercises on the Wigner-Eckart theorem (Ex. 5.8) and on the completeness relation for the SU(N) generators (Ex. 11.3) have been added. Finally, the Mathematical Sup• plement on Lie groups (Chap. 12) has been carefully checked and received a new introductory section. We thank several colleagues for helpful comments, especially Prof. L. Wilets (Seattle) for providing a list of errors and misprints. We are greatly indebted to Prof. P.O. Hess (University of Mexico) for making available corrections and valu• able material for Chap. 12. We also thank Dr. R. Mattiello who has supervised the preparation of the second edition of the book. Finally we acknowledge the agreeable collaboration with Dr. H.J. Ktilsch and his team at Springer-Verlag, Heidelberg.

Frankfurt am Main and Durham, NC, USA Walter Greiner July 1994 Berndt Miiller

Preface to the Third Printing

We are pleased to note that our Quantum Mechanics: Symmetries has found many friends among physics students and researchers so that the need for a third printing has arisen. We have taken this opportunity to make amendments and improvements to the text. A number of misprints and minor errors have been cor• rected and explanatory remarks have been given at various places. Several new examples and exercises and, in particular, a short chapter (Chap. 16) on the proof of Racah's theorem have been added. VIII Preface to the Third Printing

We thank several colleagues and students for helpful comments, particularly Prof. Dr. Peter Hess (Instituto de Ciencias Nucleares, UNAM, Mexico City) and Dr. Ulrich Eichmann who helped to improve some exercises and examples and to prepare this third printing. Finally we acknowledge the agreeable collaboration with Dr. H.J. KO!sch and his team at Springer-Verlag, Heidelberg.

Frankfurt am Main Walter Greiner July 2001 Preface to the First Edition

Theoretical physics has become a many-faceted science. For the young student it is difficult enough to cope with the overwhelming amount of new scientific ma• terial that has to be learned, let alone obtain an overview of the entire field, which ranges from mechanics through electrodynamics, quantum mechanics, field the• ory, nuclear and heavy-ion science, statistical mechanics, thermodynamics, and solid-state theory to elementary-particle physics. And this knowledge should be acquired in just 8-10 semesters, during which, in addition, a Diploma or Mas• ter's thesis has to be worked on or examinations prepared for. All this can be achieved only if the university teachers help to introduce the student to the new disciplines as early on as possible, in order to create interest and excitement that in turn set free essential, new energy. Naturally, all inessential material must simply be eliminated. At the Johann Wolfgang Goethe University in Frankfurt we therefore con• front the student with theoretical physics immediately, in the first semester. Theoretical Mechanics I and II, Electrodynamics, and Quantum Mechanics I - An Introduction are the basic courses during the first two years. These lectures are supplemented with many mathematical explanations and much support ma• terial. After the fourth semester of studies, graduate work begins, and Quantum Mechanics II - Symmetries, Statistical Mechanics and Thermodynamics, Rela• tivistic Quantum Mechanics, Quantum Electrodynamics, the Gauge Theory of Weak Interactions, and Quantum Chromodynamics are obligatory. Apart from these, a number of supplementary courses on special topics are offered, such as Hydrodynamics, Classical Field Theory, Special and General Relativity, Many• Body Theories, Nuclear Models, Models and Elementary Particles, and Solid• State Theory. Some of them, for example the two-semester courses Theoretical Nuclear Physics and Theoretical Solid-State Physics, are also obligatory. The form of the lectures that comprise Quantum Mechanics- Symmetries follows that of all the others: together with a broad presentation of the necessary mathematical tools, many examples and exercises are worked through. We try to offer science in a way as interesting as possible. With symmetries in quantum mechanics we are dealing with a particularly beautiful theme. The selected mate• rial is perhaps unconventional, but corresponds, in our opinion, to the importance of this field in modem physics. After a short reminder of some symmetries in classical mechanics, the great importance of symmetries in quantum mechanics is outlined. In particular, the consequences of rotational symmetry are described in detail, and we are soon led to the general theory of Lie groups. The isospin group, hypercharge, and X Preface to the First Edition

SU(3) symmetry and its application in modem elementary-particle physics are broadly outlined. Essential mathematical theorems are first quoted without proof and heuristically illustrated to show their importance and meaning. The proof can then be found in detailed examples and worked-out exercises. A mathematical supplement on root vectors and classical Lie algebras deep• ens the material, the Young-tableaux technique is broadly outlined, and, by way of a chapter on group characters and another on charm, we lead up to very mod• em questions of physics. Chapters on special discrete symmetries and dynamical symmetries round off these lectures. These are all themes which fascinate young physicists, because they show them that as early as the fifth semester they can properly address and discuss questions of frontier research. Many students and collaborators have helped during the years to work out ex• amples and exercises. For this first English edition we enjoyed the help of Maria Berenguer, Snjezana Butorac, Christian Derreth, Dr. Klaus Geiger, Dr. Matthias Grabiak, Carsten Greiner, Christoph Hartnack, Dr. Richard Herrmann, Raf• faele Mattiello, Dieter Neubauer, Jochen Rau, Wolfgang Renner, Dirk Rischke, Thomas SchOnfeld, and Dr. Stefan Schramm. Miss. Astrid Steidi drew the graphs and prepared the figures. To all of them we express our sincere thanks. We are also grateful to Dr. K. Langanke and Mr. R. Konning of the Physics Department of the University in Munster for their valuable comments on the German edition. We would especially like to thank Mr. Bela Waldhauser, Dipl.-Phys., for his overall assistance. His organizational talent and his advice in technical matters are very much appreciated. Finally, we wish to thank Springer-Verlag; in particular, Dr. H.-U. Daniel, for his encouragement and patience, and Mr. Michael Edmeades, for expertly copy• editing the English edition.

Frankfurt am Main Walter Greiner July 1989 Berndt Muller Contents

1. Symmetries in Quantum Mechanics ...... 1 1.1 Symmetries in Classical Physics ...... 1 1.2 Spatial Translations in Quantum Mechanics ...... 19 1.3 The Unitary Translation Operator ...... 20 1.4 The Equation of Motion for States Shifted in Space ...... 22 1.5 Symmetry and Degeneracy of States ...... 23 1.6 Time Displacements in Quantum Mechanics ...... 32 1.7 Mathematical Supplement: Definition of a Group ...... 34 1.8 Mathematical Supplement: Rotations and their Group Theoretical Properties ...... 36 1.9 An Isomorphism of the Rotation Group ...... 40 1.9.1 Infinitesimal and Finite Rotations ...... 41 1.9.2 Isotropy of Space ...... 43 1.10 The Rotation Operator for Many-Particle States ...... 54 1.11 Biographical Notes ...... 55 2. Angular Momentum Algebra Representation of Angular Momentum Operators: Generators of S0(3) ...... 57 2.1 Irreducible Representations of the Rotation Group ...... 57 2.2 Matrix Representations of Angular Momentum Operators . . . 62 2.3 Addition of Two Angular Momenta ...... 71 2.4 Evaluation of Clebsch-Gordan Coefficients ...... 75 2.5 Recursion Relations for Clebsch-Gordan Coefficients ...... 76 2.6 Explicit Calculation of Clebsch-Gordan Coefficients ...... 78 2.7 Biographical Notes ...... 85 3. Mathematical Supplement: Fundamental Properties of Lie Groups 87 3.1 General Structure of Lie Groups ...... 87 3.2 Interpretation of Commutators as Generalized Vector Products, Lie's Theorem, Rank of Lie Group ...... 98 3.3 Invariant Subgroups, Simple and Semisimple Lie Groups, Ideals ...... 100 3.4 Compact Lie Groups and Lie Algebras ...... 108 3.5 Invariant Operators (Casimir Operators) ...... 108 3.6 Theorem of Racah ...... 109 3.7 Comments on Multiplets ...... 109 3.8 Invariance Under a Symmetry Group ...... 112 XII Contents

3.9 Construction of the Invariant Operators ...... 115 3.10 Remark on Casimir Operators of Abelian Lie Groups ...... 118 3.11 Completeness Relation for Casimir Operators ...... 118 3.12 Review of Some Groups and Their Properties ...... 120 3.13 The Connection Between Coordinate Transformations and Transformations of Functions ...... 121 3.14 Biographical Notes ...... 134 4. Symmetry Groups and Their Physical Meaning: General Considerations ...... 135 4.1 Biographical Notes ...... 140 5. The Isospin Group (Isobaric Spin) ...... 141 5.1 Isospin Operators for a Multi-Nucleon System ...... 147 5.2 General Properties of Representations of a Lie Algebra ..... 155 5.3 Regular (or Adjoint) Representation of a Lie Algebra ...... 157 5.4 Transformation Law for Isospin Vectors ...... 161 5.5 Experimental Test of Isospin lnvariance ...... 169 5.6 Biographical Notes ...... 185 6. The Hypercharge ...... 187 6.1 Biographical Notes ...... 193 7. The SU(3) Symmetry ...... 195 7.1 The Groups U(n) and SU(n) ...... 195 7.1.1 The Generators ofU(n) and SU(n) ...... 198 7.2 The Generators of SU(3) ...... 200 7.3 The Lie Algebra of SU(3) ...... 202 7.4 The Subalgebras of the SU (3) Lie Algebra and the Shift Operators ...... 212 7.5 Coupling ofT, U and V Multiplets ...... 215 7.6 Quantitative Analysis of Our Reasoning ...... 216 7.7 Further Remarks About the Geometric Form of an SU(3) Multiplet ...... 218 7.8 The Number of States on Mesh Points on Inner Shells ...... 219 8. Quarks and SU (3) ...... 231 8.1 Searching for Quarks ...... 234 8.2 The Transformation Properties of Quark States ...... 234 8.3 Construction of all SU(3) Multiplets from the Elementary Representations [3] and [3] ...... 241 8.4 Construction of the Representation D(p, q) from Quarks and Antiquarks ...... 242 8.4.1 The Smallest SU (3) Representations ...... 245 8.5 Meson Multiplets ...... 247 8.6 Rules for the Reduction of Direct Products of SU(3) Multiplets ...... 258 8.7 U-Spin Invariance ...... 263 8.8 Test of U -Spin Invariance ...... 265 Contents XIII

8.9 The Gell-Mann-Okubo Mass Formula ...... 267 8.10 The Clebsch-Gordan Coefficients of the SU(3) ...... 269 8.11 Quark Models with Inner Degrees of Freedom ...... 272 8.12 The Mass Formula in SU(6) ...... 301 8.13 Magnetic Moments in the Quark Model ...... 301 8.14 Excited Meson and Baryon States ...... 304 8.14.1 Combinations of More Than Three Quarks ...... 304 8.15 Excited States with Orbital Angular Momentum ...... 306 9. Representations of the Permutation Group and Young Tableaux . 309 9.1 The Permutation Group and Identical Particles ...... 309 9.2 The Standard Form of Young Diagrams ...... 313 9.3 Standard Form and Dimension of Irreducible Representations of the Permutation Group SN ...... 316 9.4 The Connection Between SU(2) and S2 ...... 326 9.5 The Irreducible Representations of SU(n) ...... 329 9.6 Determination of the Dimension ...... 335 9.7 The SU(n -1) Subgroups of SU(n) ...... 339 9.8 Decomposition of the Tensor Product of Two Multiplets .... 341

10. Mathematical Excursion. Group Characters ...... 345 10.1 Definition of Group Characters ...... 345 10.2 Schur's Lemmas ...... 346 10.2.1 Schur's First Lemma ...... 346 10.2.2 Schur's Second Lemma ...... 346 10.3 Orthogonality Relations of Representations and Discrete Groups ...... 347 10.4 Equivalence Classes ...... 349 10.5 Orthogonality Relations of the Group Characters for Discrete Groups and Other Relations ...... 352 10.6 Orthogonality Relations of the Group Characters for the Example of the Group D(3) ...... 353 10.7 Reduction of a Representation ...... 355 10.8 Criterion for Irreducibility ...... 355 10.9 Direct Product of Representations ...... 356 10.10 Extension to Continuous, Compact Groups ...... 357 10.11 Mathematical Excursion: Group Integration ...... 357 10.12 Unitary Groups ...... 360 10.13 The Transition from U(N) to SU(N) for the example SU(3) . 361 10.14 Integration over Unitary Groups ...... 363 10.15 Group Characters of Unitary Groups ...... 366 11. Charm and SU(4) ...... 385 11.1 Particles with Charm and the SU(4) ...... 387 11.2 The Group Properties of SU(4) ...... 387 11.3 Tables of the Structure Constants fijk and the Coefficients d;jk for SU(N) ...... 397 11.4 Multiplet Structure of SU(4) ...... 399 XIV Contents

11.5 Advanced Considerations ...... 406 11.5 .1 Decay of Mesons with Hidden Charm ...... 406 11.5.2 Decay of Mesons with Open Charm ...... 407 11.5.3 Baryon Multiplets ...... 408 11.6 The Potential Model of Charmonium ...... 418 11.7 The SU(4) [SU(8)] Mass Formula ...... 426 11.8 The I Resonances ...... 430

12. Mathematical Supplement ...... 435 12.1 Introduction ...... 435 12.2 Root Vectors and Classical Lie Algebras ...... 439 12.3 Scalar Products of Eigenvalues ...... 444 12.4 Cartan-Weyl Normalization ...... 447 12.5 Graphical Representation of the Root Vectors ...... 448 12.6 Lie Algebra of Rank l ...... 449 12.7 Lie Algebras of Rank 2 ...... 449 12.8 Lie Algebras of Rimk l > 2 ...... 450 12.9 The Exceptional Lie Algebras ...... 451 12.10 Simple Roots and Dynkin Diagrams ...... 452 12.11 Dynkin' s Prescription ...... 454 12.12 The Cartan Matrix ...... 456 12.13 Determination of all Roots from the Simple Roots ...... 457 12.14 Two Simple Lie Algebras ...... 459 12.15 Representations of the Classical Lie Algebras ...... 460

13. Special Discrete Symmetries ...... 465 13.1 Space Reflection (Parity Transformation) ...... 465 13.2 Reflected States and Operators ...... 467 13.3 Time Reversal ...... 468 13.4 Antiunitary Operators ...... 470 13.5 Many-Particle Systems ...... 474 13.6 Real Eigenfunctions ...... 475 14. Dynamical Symmetries ...... 477 14.1 The Hydrogen Atom ...... 477 14.2 The Group S0(4) ...... 480 14.3 The Energy Levels of the Hydrogen Atom ...... 481 14.4 The Classical Isotropic Oscillator ...... 482 14.4.1 The Quantum Mechanical Isotropic Oscillator ..... 483

15. Mathematical Excursion: Non-compact Lie Groups ...... 497 15.1 Definition and Examples of Non-compact Lie Groups ...... 497 15.2 The Lie Group S0(2, 1) ...... 505 15.3 Application to Scattering Problems ...... 509

16. Proof of Racah's Theorem ...... 513 Subject Index ...... 519 Contents of Examples and Exercises

1.1 Angular Momenta in Different Reference Frames ...... 5 1.2 Conserved Quantities of Specified Fields ...... 6 1.3 Noether's Theorem (for Improved Insight) ...... 8 1.4 Time-Invariant Equations of Motion: The Lagrange Function and Conserved Quantities ...... 11 1.5 Conditions for Translational, Rotational and Galilean Invariance . 12 1.6 Conservation Laws in Homogeneous Electromagnetic Fields . . . . 16 1.7 Matrix Elements of Spatially Displaced States ...... 25 1.8 The Relation (ifJ/n)n B(x) and Transformation Operators ...... 26 1.9 Translation of an Operator A(x) ...... 28 1.10 Generators for Translations in a Homogeneous Field ...... 29 1.11 Transformation of Vector Fields Under Rotations ...... 44 1.12 Transformation of Two-Component Spinors Under Rotations . . . . 48 1.13 Measuring the Direction of Electron Spins ...... 52 2.1 Special Representation of the Spin-1 Operators ...... 64 2.2 Calculation of the Clebsch-Gordan Coefficients for Spin-Orbit Coupling ...... 82 3.1 Lie Algebra of S0(3) ...... 90 3.2 Calculation with Complex n x n Matrices ...... 91 3.3 Proof of a Commutation Relation ...... 93 3.4 Generators and Structure Constants of Proper Lorentz Transformations ...... 94 3.5 Algebra of Pv and fv ...... 99 3.6 Translation-Rotation Group ...... 100 3.7 Simple_ and Semisimf.le Li~ Groups ...... 102 3.8 ReductiOn of exp{- 211/m · u} ...... 104 3.9 Cartan's Criterion for Semisimplicity ...... 105 3.10 Semisimplicity of S0(3) ...... 107 3.11 An Invariant Subspace to the Rotation Group ...... 109 3.12 Reduction of an Invariant Subspace ...... 110 3.13 Casimir Operator of the Rotation Group ...... 113 3.14 Some Groups with Rank 1 or 2 ...... 114 3.15 Construction of the Hamiltonian from the Casimir Operators . . . . 119 3.16 Transformations with r Parameters of ann-Dimensional Space .. 123 3.17 Generators and Infinitesimal Operators of SO(n) ...... 124 3.18 Matrix Representation for the Lie Algebra of Spin l ...... 126 XVI Contents of Examples and Exercises

3.19 Translations in One-dimensional Space; the Euclidean Group E(3) in Three Dimensions ...... 130 3.20 Homomorphism and Isomorphism of Groups and Algebra ...... 132 3.21 Transformations of the Structure Constants ...... 133 4.1 Conservation Laws with Rotation Symmetry and Charge Independent Forces ...... 136 4.2 Energy Degeneracy for Various Symmetries ...... 139 4.3 Degeneracy and Parity of More Symmetries ...... 140 5.1 Addition Law for Infinitesimal SU(2) Transformations ...... 145 5.2 The Deuteron ...... 149 5.3 The Charge Independence of Nuclear Forces ...... 151 5.4 The Pion Triplet ...... 152 5.5 Normalization of the Group Generators ...... 158 5.6 The G Parity ...... 164 5.7 Representation of a Lie Algebra, Regular Representation of the Algebra of Orbital Angular Momentum Operators ...... 168 5.8 The Wigner-Eckart Theorem ...... 172 5.9 Pion Production in Proton-Deuteron Scattering ...... 175 5.10 Production of Neutral Pions in Deuteron-Deuteron Scattering ... 176 5.11 Pion-Nucleon Scattering ...... 176 5.12 The Decay of the Neutral Rho Meson ...... 183 6.1 Hypercharge of Nuclei ...... 188 6.2 The Hypercharge of the~ Resonances ...... 189 6.3 The Baryons ...... 189 6.4 Antibaryons ...... 191 6.5 Isospin and Hypercharge of Baryon Resonances ...... 191 7.1 The Lie Algebra of SU(2) ...... 199 7.2 Linear Independence of the Generators 5..; ...... 20 I 7.3 Symmetry of the Coefficients d;Jk ...... 204 7.4 Antisymmetry of the Structure Constants fiJk ...... 206 7.5 Calculation of some dijk Coefficients and Structure Constants .... 207 7.6 Relations Between the Structure Constants and the Coefficients d;Jk ...... 209 7.7 Casimir Operators of SU(3) ...... 210 7.8 Useful Relations for SU(3) Casimir Operators ...... 211 7.9 The Increase of the Multiplicity of States on the Inner Shells of SU(3) Multiplets ...... 221 7.10 Particle States at the Centre of the Baryon Octet ...... 225 7.11 Calculation of the Dimension of the Representation D(p, q) ..... 226 7.12 The Casimir Operator C1 = L; f? in terms of p and q ...... 228 8.1 The Generators of SU(3) in the Representation [3] ...... 235 8.2 Transformation Properties of the States of the Anti triplet [3] .... 238 8.3 Non-equivalence of the Two Fundamental Representations of SU(3) ...... 239 8.4 The Weight of a State ...... 243 Contents of Examples and Exercises XVII

8.5 The Maximum Weight of the Quark Triplet [3] and Antiquark Triplet (3] ...... 244 8.6 The Pseudoscalar Mesons ...... 248 8.7 Example (for Deeper Insight): The K0 and K0 Mesons and Their Decays ...... 249 8.8 The Scalar Mesons ...... 255 8.9 The Vector Mesons ...... 256 8.10 The Tensor Mesons ...... 257 8.11 Other Resonances ...... 258 8.12 Reduction of SU(2) Multiplets ...... 261 8.13 Construction of the Neutron Wave Function ...... 280 8.14 Construction of the Wave Functions of the Baryon Decuplet .... 283 8.15 Construction of the Spin-Flavour Wave Functions of the Baryon Octet ...... 292

9.1 Basis Functions of S3 ...... 318 9.2 Irreducible Representations of S4 ...... 320 9.3 Multiplets of a System of Three Spin-! Particles ...... 327 9.4 Multiplets of a Two-Particle System in the Group SU(3) ...... 330 9.5 Multiplets of the SU(3) Constructed from Three Particles ...... 331 9.6 Dimension Formula for the SU(3) ...... 337 9.7 Decomposition of a Tensor Product ...... 343 9.8 Representations of the SU(2) and Spin ...... 344 9.9 Triality and Quark Confinement ...... 344

10.1 The Group D3 ...... 350 10.2 The Rotation Group 0(3) ...... 351 I 0.3 Application of Group Characters: Partition Function for the Colour Singlet Quark-Gluon Plasma with Exact SU(3) Symmetry ...... 373 10.4 Proof of the Recursion Formula for the Dimensions of the SU(n) Representations ...... 378

11.1 Anticommutators of the Generators of SU (N) ...... 389 11.2 Trace of a Generator Product in the SU (N) ...... 391 11.3 The Completeness Relation for Fa ...... 392 11.4 Eigenvalue of the Casimir Operator C1 of a Fundamental Representation of the SU(N) ...... 396 11.5 SU(3) Content of the SU(4) Meson Multiplet ...... 411 11.6 Decomposition of the Product [4] 0 [4] 0 [4] ...... 413 11.7 SU(3) Content of the SU(4) Baryon Multiplet ...... 414 11.8 Decomposition and Dimension of Higher SU (4) Multiplets ..... 416 11.9 Mathematical Supplement. Airy Functions ...... 422

12.1 Weight Operators of the SU(4) Algebra ...... 438 12.2 Proof of a Relation for the Structure Constants Cikl ...... 444 12.3 Dynkin Diagrams for B1 ..•..•..•..•..•••.••••••••.•••• 455 12.4 The Cartan Matrices for SU(3), SU(4) and G2 ...... 456 XVIII Contents of Examples and Exercises

12.5 Determination of the Roots ofG2 Using the Corresponding Simple Roots ...... 458 12.6 Analysis of SU(3) ...... 463 13.1 Effect of an Antiunitary Operator on Matrix Elements of Wavefunctions ...... 470 13.2 Commutation Relations Between{; and S ...... 473 14.1 Energy and Radial Angular Momentum of the Hydrogen Atom ...... 484 14.2 The Runge-Lenz Vector ...... 485 14.3 Properties of the Runge-Lenz Vector M ...... 485 14.4 The Commutator Between M and ii ...... 486 14.5 The Scalar Product L · M ...... 489 14.6 Determination of Ml ...... 490 14.7 Proof of the Commutation Relation for [ M;, Lj l- ...... 492 14.8 Proof of the Commutation Relation for [M;, Hj l- ...... 494 15.1 Representation of SU(2) Matrices ...... 498 15.2 Representation of SU(l, 1) Matrices ...... 499 15.3 Non-compactness of the Lorentz Group ...... 501 15.4 Generators of SO(p, q) ...... 501 15.5 Casimir Operator of S0(2, 1) ...... 506 15.6 Coordinate Representation of S0(2, 1) Operators ...... 509 16.1 The Eigenvalue of the Second-Order Casimir Operator of SU(3) . 518