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Physical Review A

Expansion of the spin-spin interaction

Keh‐Ning Huang

Published Nov 1, 1980• 0 citations
Publisher

Abstract

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 $m$-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 $m$-scheme matrix element enables one to obtain graphically matrix elements between many-particle configurations in any coupling scheme.

DOI
10.1103/physreva.22.1775

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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.
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Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are reduced to the independent sets of two-component spinors defined as a product of complete orthonormal sets of radial parts of orbitals and twocomponent spinor type tensor spherical harmonics. We notice that the new idea presented in this work is the unified treatment of half-integral spin and scalar particles in position, momentum and four-dimensional spaces. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of different problems arising in the relativistic quantum mechanics when the orthonormal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.
Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.