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Matrix Elements of Tensor Operators

D.M. Brink, G.R. Satchler

Published Feb 10, 1994• 0 citations
Publisher

Abstract

Abstract WE are now in a position to calculate those parts of any physical problem which are concerned with angular integrations, the coupling of angular momentum vectors, or our choice of coordinate system-in fact general rotational symmetries-which we might call the ‘geometrical’ parts. The basic technique is to exploit the simplest matrix reduction which we have, already met in Chapter IV, namely the WignerEckart theorem for the matrix elements of spherical tensors.

DOI
10.1093/oso/9780198517597.003.0005

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Abstract WE are now in a position to calculate those parts of any physical problem which are concerned with angular integrations, the coupling of angular momentum vectors, or our choice of coordinate system-in fact general rotational symmetries-which we might call the ‘geometrical’ parts. The basic technique is to exploit the simplest matrix reduction which we have, already met in Chapter IV, namely the WignerEckart theorem for the matrix elements of spherical tensors.
Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.
This is the most complete handbook on the quantum theory of angular momentum. Containing basic definitions and theorems as well as relations, tables of formula and numerical tables which are essential for applications to many physical problems, the book is useful for specialists in nuclear and particle physics, atomic and molecular spectroscopy, plasma physics, collision and reaction theory, quantum chemistry, etc. The authors take pains to write many formulae in different coordinate systems thus providing users with added ease in consulting this book. Each chapter opens with a comprehensive list of its contents to ease the search for any information needed later. New results relating to different aspects of the angular momentum thoery are also included. Containing close to 500 pages this book also gathers together many useful formulae besides those related to angular momentum. The book also compares different notations used by previous authors.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).