In this post, we will see the book *Theory of Angular Momentum *by A. P. Levinson, I. B. Vanagas, V. V. Yutsis.

About the book

FOREWORD

The principal results obtained up to 1935 in the quantum-mechanical theory of angular momentum are contained in chapter III of Condon and Shortley’s “Theory of Atomic Spectra” /1949/. Since then, owing to the ideas of Wigner /1931,1937/ and Racah /1942/, the theory has been enriched by the algebra of noncommuting tensor operators and the theory of y-coefficients. This has considerably increased its computational possibilities and has broadened the scope of its applications. Among the branches of theoretical physics where the methods of the theory of angular momentum are widely applied today we might mention the theory of atomic and nuclear spectra, the scattering of polarized particles in nuclear reactions, the theory of genealogical coefficients, etc. (a bibliography of the applications may be found in Edmonds’ book /1957/).

The only book known to us giving an exposition of the algebra of noncommuting tensor operators and j-coefficients is Edmonds’ “Angular Momentum in Quantum Mechanics” /1957/, which may serve as an excellent textbook for a first acquaintance with the subject. However, the exposition of the theory of j-coefficients and transformation matrices given in this book is not complete. This may constitute an impediment when the apparatus is employed m more complicated cases. The present

book fills this gap.The writing of this book began before Edmonds’ book appeared in print. The authors have utilized nearly all results known to them in the given field. Among these a certain place is occupied by the results obtained by a group of workers under the

direction of one of the present authors (A. Yutsis), the remaining two authors (I. Levmson and V. Vanagas) being the principal participants. The book corresponds to the content of the first part of a course, “Methods of Quantum-Mechanical Atomic Calculations”, given by the senior author to students of theoretical physics at the

Vilnius State University im. V. Kapsukas over the last two years.We found it worthwhile to use the elegant and powerful methods of group theory in our exposition. To avoid encumbering the book with elements of group theory we have assumed that the reader is already acquainted with linear representations of the three-dimensional rotation group. The reader who is unfamiliar with this may refer to the books by G.Ya. Lyubarskii*/1957/ and I. M, Gel’fand et al. /1958/.

We begin with the well-known theory of vector addition of two angular momenta (chapter I), turning next to the addition of an arbitrary number of angular momenta (chapter II), The following chapters (III- VI) are devoted to quantities of the theory of angular momentum where an important place is occupied by the graphical method which IS convenient for various calculations. The last chapter (VII) deals with the method of noncommuting tensor operators. Material of a supplementary character is given in the appendices.

We have cited a number of unpublished works some of which were not available to us. References to these were based on other published works. We apologise in advance for any resulting inaccuracy.

The book was translated from the Russian by A. Sen and R.N. Sen and was published by Israel Program for Scientific Translations for the National Science Foundation and the National Aeronautics and Space Administration, U.S.A. in 1962.

Contents

FOREWORD

TRANSLATOR’S NOTE

Chapter I ADDITION OF TWO ANGULAR MOMENTA 1

1. Angular momentum operators and spatial rotations 1

2. Angular momentum eigenfunctions and representations of the rotation group 2

3. Addition of angular momenta, reduction of the direct product of representations of the rotation group 5

4. Expressions for the Clebsch-Gordan coefficients and their properties 8

5. Wigner coefficients and their properties 13

Chapter II ADDITION OF AN ARBITRARY NUMBER OF ANGULAR MOMENTA 16

6. General considerations on the addition of an arbitrary number of angular momenta 16

7. Group -theoretic considerations on the generalized Clebsch-Gordan coefficients 19

8. The transformation matrix 21

9. Simplification of the transformation matrix 24

10. Generalized Wigner coefficients and their properties 27

Chapter III GRAPHICAL METHODS FOR OPERATIONS WITH SUMS OF PRODUCTS OF WIGNER COEFFICIENTS 31

11. Sums of products of Wigner coefficients (** jm**-coefficients) 31

12. Graphical representation of

*-coefficients 34*

**jm**13. Expansion of

**-coefficients in generalized Wigner coefficients 39**

*jm*14. Transformation of

**-coefficients 42**

*jm*15. Summation of

**-coefficients 46**

*jm*Chapter IV ** j**-COEFFICIENTS AND THEIR PROPERTIES 49

16. The 6 ** j**-coefficient and its properties 49

17. 3n

**-coefficients of the first and second kinds 55**

*j*18. The 9

**-coefficient and Its properties 59**

*j*19. 12

**-coefficients and their properties 62**

*j*20. Methods of studying j-coefficients, 15

**-coefficients 65**

*j*Chapter V UTILIZATION OF TRANSFORMATION MATRICES FOR OBTAINING SUM RULES AND TRANSFORMATION FORMULAS FOR ** jm**-COEFFICIENTS 71

21. General considerations on the relation between transformation matrices and ** j**-coefficients 71

22. Methods for obtaining the relation between transformation matrices and

**-coefficients 74**

*j*23. Explicit expressions for the simplest transformation matrices 76

24. Utilization of matrix identities for obtaining sum rules on

**-coefficients 80**

*j*25. Use of matrix identities for the transformation of

**-coefficients 83**

*jm*Chapter VI EXAMPLES OF APPLICATION OF THE GRAPHICAL METHOD 87

26. Graphical summation of products of Wigner coefficients 87

27. A more complex product of Wigner coefficients 91

28 Summation of a product of ** j**-coefficients 94

29. Summation of a product of Wigner coefficients and j-coefficients 98

30. Choice of a method of calculation 102

Chapter VII. IRREDUCIBLE TENSOR OPERATORS AND EXPRESSIONS FOR THEIR MATRIX ELEMENTS 105

31. Irreducible tensor operators and their properties 105

32. Tensor products 107

33. Expressions for matrix elements of products of tensor operators 109

34. Calculation of matrix elements of complex products of tensor operators 111

35. Double tensors, their products and matrix elements 114

Appendix 1. NOTATIONS FOR THE WIGNER, 6** j**-, 9

**– AND ALLIED COEFFICIENTS 117**

*j*Appendix 2. ALGEBRAIC FORMULAS FOR THE CLEBSCH-GORDAN COEFFICIENTS 119

Appendix 3. DIAGRAMS OF 18 ** j**-COEFFICIENTS 122

Appendix 4. PROPERTIES OF 18 ** j**-COEFFICIENTS 126

Appendix 5. EXPRESSIONS FOR THE TRANSFORMATION MATRICES OF EIGENFUNCTIONS OF FIVE COUPLED ANGULAR MOMENTA 135

Appendix 6. SUM RULES ON j-COEFFICIENTS 141

Appendix 7 THE SIMPLEST SUMMATION AND TRANSFORMATION FORMULAS FOR ** jm**-COEFFICIENTS 152

BIBLIOGRAPHY 154

SUPPLEMENTARY BIBLIOGRAPHY 157

INDEX 158