Coupling of two angular momenta

As the angular momentum is a physical vector, two angular momenta, say /[ and l2, obey vector addition yielding an angular momentum L 

In the role of the components either the orbital angular momenta or the spin angular momenta can operate, both at the same centre or at different centres. 

It is assumed that the eigenstates of each component are known and fulfil characteristic equations such as 

The compound angular momentum should obey similar equations LZ LM) = Mh LM) 

The coefficients of the transformation depend on all six independent indices (variables) 

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The coupling of two angular momenta with equal magnimdes, as in the previous model of the perturbed H atom, is of special interest in leading to a monodromy matrix with index n = 2. 

The individual J,Mj) multiplets cannot be, in principle, described by a single Slater determinant. This is a consequence of the general rules of coupling of two angular momenta eigenfunctions of all projections of the total angular momentum which is lower than the maximal possible (L + S), are represented as linear combinations of several determinants [36]. 

For the coupling of two angular momenta a and b to the resulting value c with the corresponding magnetic quantum numbers a, P and y, the Clebsch-Gordan coefficients (atxbp cy) are defined as expansion coefficients in the relation... 

The Clebsch-Gordan coefficient is unsymmetrical between j and j2 on the one hand and j on the other. There is also an asymmetry between j and j2 which arises from the order of the coupling and leads to a different phase factor. A more symmetrical formulation of the coupling coefficients is possible, as shown by Wigner [11], Consider first the coupling of two angular momenta, both with quantum number j, to give a scalar (J = 0) result. We have... 

In order to understand formal relationships the basic coupling of two angular momenta is expressed in terms of the Clebsch—Gordan coefficients. In setting up the algebra for computation it is more convenient to use the 3-j symbols. Calculations of the Wigner symbols and the Clebsch—Gordan coefficients are found in subroutine libraries. See, for example, Soper (1989). 

Therefore the transformation matrix C which diagonalises the spin-orbit coupling operator can be generated by the well-known techniques for coupling of two angular momenta, one of them having S = 1. 

Clebsch-Gordan coefficients and Wigner s (y symbols coupling of two angular momenta... 

We now consider the coupling of two angular momenta. For this purpose we have to specify Equation [31] to the case G = SU(2) which is here written as ... 

The simplest example of coupled algebras is provided by the angular momenta (cf. note 2 of Chapter 1). The direct sum of two angular momenta, J and J2, is just... 

The addition of two angular momenta (formula of the type (10.4)) may be directly generalized to cover the case of an arbitrary number of momenta. However, in such a case it is not enough to adopt the total momentum and its projection for the complete characterization of the wave function of coupled momenta. Normally, the quantum numbers of intermediate momenta must be exploited too. Moreover, these functions depend on the form (order) of the coupling between these momenta. The relationships between the functions, belonging to different forms of coupling of their momenta, may be found with the aid of transformation matrices. 

The wave function for the particular case of two equivalent electrons may be constructed, using vectorial coupling of the angular momenta and antisymmetrization procedure. For LS and jj coupling, it will look as... 

The coupling of four angular momenta again can be performed by different routes using a different set of intermediate spins. For example, we can select two coupling schemes ... 

Depending on the individual orbital angular momenta /a and tb involved, the summation can go up to infinity. Such a situation occurs for double photoionization in helium where the 1P° state of the continuum pair wavefunction can be obtained by an unlimited coupling of individual orbital momenta (esep, sped, edef,...). However, in the case of photon-induced two-step double ionization the formulation... 

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