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Physics of Atomic Nuclei

Simple numerical method of computing the probabilities of angular momentum J ν occurring among possible vectors J Resulting from n angular momentum j μ summation (μ=1−n)

Maria Kaczmarczyk

Published Nov 1, 2004• 1 citation
Publisher

Abstract

Arelatively simple numerical method of summing angular momentum vectors with maintaining space quantization rules of each summed angular momentum has been presented. The method enables the calculation of the values of probability ( p j μ ) of finding a definite angular momentum J μ among all vectors J being the results of quantum summation of n angular momentum vectors j μ (μ=1- n . It may be used, e.g., in the calculations of angular momentum of many-particle states. The significance of the paper is connected with the possibility of taking into account, in a simple way, the angular momentum conservation principle for a system which consists of an arbitrary number of excitons.

DOI
10.1134/1.1825535

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Arelatively simple numerical method of summing angular momentum vectors with maintaining space quantization rules of each summed angular momentum has been presented. The method enables the calculation of the values of probability ( p j μ ) of finding a definite angular momentum J μ among all vectors J being the results of quantum summation of n angular momentum vectors j μ (μ=1- n . It may be used, e.g., in the calculations of angular momentum of many-particle states. The significance of the paper is connected with the possibility of taking into account, in a simple way, the angular momentum conservation principle for a system which consists of an arbitrary number of excitons.
Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
This book is concerned with the practical aspects of solving angular momentum problems. The novel but fully tested-out method (the Invariant Graph Method) allows one to write down from a single graph the complete final result of the problem. The drawing of the graph involves very few simple, essentially self-evident rules. Still it is a powerful tool to easily solve the most involved physical problems. The method is introduced step-by-step in a sequence of examples, beginning with the simplest matrix elements, and ending with the most general case of a reaction including angular distributions and correlations. The many-body and particle anti-particle systems are fully developed. All aspects: wave functions, vectors, operators, Fock space state vectors and operators, etc., are treated on the same footing. All concepts of angular momentum theory acquire a transparent meaning. Hence the book is valuable not only as a handbook in problem solving, but extremely so as an adjunct in any course on advanced qunatum physics, atomic, molecular, nuclear and particle physics
Anisotropy of angular momenta: ideas and methods The fundamental idea forming the basis of early quantum theory was the quantization of the angular momentum of the atom, which, at a later stage, assumed the form of the Bohr–Sommerfeld quantization rule [360], namely ∲ pdq = n ℏ where p and q are the generalized momentum and coordinate. This concept has retained its physical meaning in modern-day quantum mechanics [371, 372]. In the course of the further development of quantum mechanical concepts it became increasingly clear that in the theory of atoms and molecules the angular momentum is a ‘dominant’ notion, the fundamental significance of which is in no small degree connected with its dimension of action, which coincides with that of Planck's constant ℏ. In the interaction between particles and a directed beam of photons the angular momentum of the system is conserved, thus forming the basis for the optical method of producing spatial anisotropy of the distribution of the angular momenta of the ensemble of particles, i.e. its optical polarization . In terms of quantum concepts, spatial anisotropy means that in an ensemble of particles a non-equilibrium population of magnetic sublevels m j of angular momenta is created. We make a distinction between the alignment of momenta, for instance in the absorption of linear polarized light, when there is no difference between the population of ± m j -sublevels, and orientation , when such a difference exists. Anisotropic distribution of angular momenta in an excited state, as produced in absorption, manifests itself directly in the polarization of fluorescence. It is rather likely that understanding of the anisotropic distribution of the angular momenta of atoms was enhanced by Hanle's [184] discovery in 1924.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.
One of the most important applications of the quantum mechanical formalism developed in the previous chapter is to angular momentum. This is because many microsystems are spherically symmetric, or approximately so, and angular momentum is therefore conserved. This chapter discusses that a careful study of quantized angular momentum leads to a completely new physical phenomenon – intrinsic spin. The fact that the angular momentum vector of a quantum particle must point in certain restricted directions is a bizarre quantum property. In some respects the abstract idea of an internal angular momentum is reminiscent of the classical concept of a spinning body. In classical mechanics angular momentum is added vectorially but in the quantum case the issue is more subtle. The procedure will be illustrated by the example of the addition of two spins. As in classical mechanics a charged quantum system with angular momentum possesses a magnetic moment μ.