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Addition theorems for spin spherical harmonics. I Preliminaries

Antonio O. Bouzas

Published Jan 1, 2013• 4 citations
OpenAlex

Abstract

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.

Related papers

Addition theorems for spin spherical harmonics: I. Preliminaries

Antonio O Bouzas

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 ).

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Generalized operator equivalents to the (bipolar) spherical harmonics [Formula: see text] are considered, viz. [Formula: see text], where the PJ are projection operators on the manifolds of definite angular momenta, A closed formula is derived for the coefficients of the Clebsch–Gordan decomposition of products of such operators.

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The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.

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Matrix Elements of Tensor Operators

D.M. Brink, G.R. Satchler

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.

•Feb 10, 1994•0 citations
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 ).
The space part of a general spin-spin interaction is expanded in a complete orthonormal set of irreducible tensors constructed from spherical harmonics of the interacting particles. With this expansions, the mmm-scheme matrix element of the spin-spin interaction is represented as a sum of weighted radial integrals along with its graphical representation. The graphical form of the mmm-scheme matrix element enables one to obtain graphically matrix elements between many-particle configurations in any coupling scheme.
The addition theorem for spherical harmonics is just a special case of the law for the transformation of spherical harmonics under rotation of their arguments. Certain operators are scalars under rotation; this means that they commute with the total angular momentum and are unchanged by the transformation. The types of operators having simple transformation properties under rotation are known generally as tensor operators. The energy eigenstates of an atom, taking into account the spin-orbit interaction, are not usually eigenstates of the z component of the total orbital angular momentum, but are eigenstates of the total angular momentum, spin plus orbital. Sehwinger has looked at the theory of angular momentum from a point of view that enables one to carry out very elegant calculations of the various quantities occurring in the theory of angular momentum. The quanta of some different oscillators are simply the quarks that compose the strongly interacting particles.
The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.