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Journal of Physics A Mathematical Nuclear and General

Trace techniques for angular momentum operators

P. R. Subramanian, Varadarajan Devanathan

Published Nov 1, 1974• 14 citations
Publisher

Abstract

General expressions for traces of products of angular momentum matrices (both in Cartesian and spherical bases) for systems of arbitrary spin are derived using the well established properties. It is shown that the trace is a polynominal in eta where eta is the eigenvalue of the J 2 operator. The trace techniques developed herein are applied to the problems of spin orientation.

DOI
10.1088/0305-4470/7/16/004

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Physical Review A•Mar 1, 1980•1 citation
General expressions for traces of products of angular momentum matrices (both in Cartesian and spherical bases) for systems of arbitrary spin are derived using the well established properties. It is shown that the trace is a polynominal in eta where eta is the eigenvalue of the J 2 operator. The trace techniques developed herein are applied to the problems of spin orientation.
Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics which are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
Angular momentum is essential to any advanced discussion of magnetic resonance spectroscopy. A brief history of the modern sub-atomic particle-spin concept is presented. The origin of the spin-1/2 angular momentum quantum number is detailed and angular momentum operators are constructed for spin-1/2 nuclei using the postulates of quantum mechanics that are then used to produce a set of very useful eigenvalue formulas. Angular momentum operators in spherical coordinate form are developed and operator matrices are constructed from Cartesian operators. The time-evolution of these operators is of much interest in later chapters and is explored in some detail here.
Synopsis: General operator properties of angular momenta are recalled, and a physical motivation for adding them is given. The product and total angular momentum bases are defined. It is shown how to express one basis in terms of the other, and an explicit example of adding two spin 1/2 angular momenta is given. A further example consists of adding a general orbital angular momentum value to spin 1/2. These considerations are applied to perturbations in the hydrogen atom energy spectrum, where the correct choice for calculations is the total angular momentum basis.