Angular Momentum I: Basics

James D. Louck

THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.

Michel Le Bellac

A. J. van Wonderen, K. Lendi

The standard representation of angular momentum on Bargmann's Hilbert space of analytic functions is extended such that domains of operators include non-analytic functions, which are square integrable. Generalized eigenfunctions of angular momentum are proposed, which contain arbitrary complex constants. The eigenfunctions are shown to satisfy a basic requirement which follows from group theory.

P. C. W. Davies, David S. Betts

D.M. Brink, G.R. Satchler

Abstract This book introduces the quantum theory of angular momentum to students who are unfamiliar with it and develops it to a stage useful for research. The first part contains the basic theory of rotations and angular momentum. As the book aims to emphasize applications, mathematical details are avoided and difficult theorems stated without proof. The second part contains examples of applications to a wide range of physical phenomena and presents a collection of results helpful in solving problems.