Abstract
Angular momentum theory is presented from the viewpoint of the group SU(1) of unimodular unitary matrices of order 2. This is the basic quantum mechanical rotation group for implementing the consequences of rotational symmetry into isolated complex physical systems and gives the structure of the angular momentum multiplets of such systems. This entails the study of representation functions of SU(2), the Lie algebra of SU(2) and copies thereof, and the associated Wigner–Clebsch–Gordan coefficients, Racah coefficients, and 3n − j coefficients, with an almost boundless set of interrelations, and presentations of the associated conceptual framework. The relationship of SU(2) to the rotation group in physical 3-space R3 is given in detail. Formulas are often given in a compendium format with brief introductions on their physical and mathematical content. A special effort is made to interrelate the material to the special functions of mathematics and to the combinatorial foundations of the subject.
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Acknowledgements
This contribution on angular momentum theory is dedicated to Lawrence C. Biedenharn, whose tireless and continuing efforts in bringing understanding and structure to this complex subject is everywhere imprinted.
We also wish to acknowledge the many contributions of H. W. Galbraith and W. Y. C. Chen in sorting out the significance of results found in Schwinger 3 . The Supplement is dedicated to the memory of Brian G. Wybourne, whose contributions to symmetry techniques and angular momentum theory, both abstract and applied to physical systems, was monumental.
The author expresses his gratitude to Debi Erpenbeck, whose artful mastery of TeX and scrupulous attention to detail allowed the numerous complex relations to be displayed in two-column format.
Author's note. It is quite impossible to attribute credits fairly in this subject because of its diverse origins across all areas of physics, chemistry, and mathematics. Any attempt to do so would likely be as misleading as it is informative. Most of the material is rooted in the very foundations of quantum theory itself, and the physical problems it addresses, making it still more difficult to assess unambiguous credit of ideas. Pragmatically, there is also the problem of confidence in the detailed correctness of complicated relationships, which prejudices one to cite those relationships personally checked. This accounts for the heavy use of formulas from 1 , which is, by far, the most often used source. But most of that material itself is derived from other primary sources, and an inadequate attempt was made there to indicate the broad base of origins. While one might expect to find in a reference book a comprehensive list of credits for most of the formulas, it has been necessary to weigh the relative merits of presenting a mature subject from a viewpoint of conceptual unity versus credits for individual contributions. The first position was adopted. Nonetheless, there is an obligation to indicate the origins of a subject, noting those works that have been most influential in its developments. The list of textbooks and seminal articles given in the references is intended to serve this purpose, however inadequately.
Excerpts and Fig. 2.1 are reprinted from Biedenharn and Louck 1 with permission of Cambridge University Press. Tables 2.2–2.4 have been adapted from Edmonds 18 by permission of Princeton University Press. Thanks are given for this cooperation.
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Louck, J.D. (2023). Angular Momentum Theory. In: Drake, G.W.F. (eds) Springer Handbook of Atomic, Molecular, and Optical Physics. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-030-73893-8_2
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