Transformation Clebsch-Gordan coefficients

Formula (3) of the Introduction was the first example of how to use transformation matrices. The simplest case of transformation matrices is presented by formulas (10.4) and (10.5), while considering the relationships between the wave functions of coupled and uncoupled momenta. The Clebsch-Gordan coefficients served there as the corresponding transformation matrices. However, the theory of transformation matrices is most widely utilized for transformations of the wave functions and matrix elements from one coupling scheme to another (Chapters 11,12) as well as for their calculations. 

The theory of transformation matrices is described in detail in [9]. In [11] universal and efficient graphical methods of operating with Clebsch-Gordan coefficients and transformation matrices are described. Their use allows one easily to find expressions for the most complex matrix elements of the operators, corresponding to physical quantities, in various coupling schemes, and to carry out their processing if necessary. Here we shall present only the minimal data on transformation matrices and methods of their evaluation. 

The transformation matrix may be expressed in terms of the product of Clebsch-Gordan coefficients, summed over all projection parameters, in other words, in terms of 3n./-coefficients. Such sums are real numbers, therefore the transformation matrices will be real as well, i.e. 

Performing the summation of the products of Clebsch-Gordan coefficients over all projection parameters, we obtain the invariants, called 3n/-coefficients. Their trivial cases were presented in Figs 8.6b-e (Kro-necker and triangular deltas). The first non-trivial case is represented by formula (6.16) or the transformation matrix of three angular momenta, connected with 6/-coefficient (see formula (6.42) and Fig. 8.7)). 

The symmetry of a number of atomic quantities (wave functions, matrix elements, 3n./-coefficients etc.) with respect to certain substitution groups or simply substitutions like l — — / — 1, L — —L — 1, S — — S — 1, j — — j— 1, N 41 + 2 — N, v —y 41 + 4 — v leads to new expressions or helps to check already existing formulas or algebraic tables [322, 323]. Some expressions are invariant under such transformations. For example, Eq. (5.40) is invariant with respect to substitutions S — —S — 1 and v — 41 + 4 — v. Clebsch-Gordan coefficients in Table 7.2 are invariant under transformation j — —j — 1. However, applying this substitution to the coefficient in Table 7.2, we obtain the algebraic value of the other coefficient. 

Ayl transforms as an irreducible tensor operator under operations of G, and as a rank-2 spinor in the angular momentum algebra generated by the quasispin operators. We form the quasispin generators as a coupled tensor in quasispin space Q(A) = i[AAAA]7V2, where [AB] = Y.qq lm q c/)AqBqi. In the Condon and Shortley spherical basis choice (with m = 1, 0, — 1) for the SO(3) Clebsch-Gordan coefficients [11-13,21-23] this takes the form [6,21] ... 

The Clebsch Gordan coefficients for given values of y) and j2 form a square matrix labelled by the j, m values one way and by mu m2 the other. This matrix is always real and orthogonal, so that the inverse transformation to (5.77) is... 

For some hand calculations, however, it is advantageous to have the relevant Clebsch-Gordan coefficients or the 3/-symbols at one s disposal (Appendix 3). Notice that the transformation of the local basis to the compound basis is represented by a unitary (orthogonal) matrix... 

The second transformation matrix U(123) is an expansion of the matrix of the previous generation U(l2) and Clebsch-Gordan coefficients for a new set of... 

Just this matrix is generated from the previous generation of the coupling matrix U(12) and the new set of Clebsch-Gordan coefficients listed in Table 1.12. The resulting transformation matrix is... 

The coupling of the local microstates into the proper molecular states is provided by the diagonalisation of the spin Hamiltonian matrix. For the zero-field case the diagonalisation matrix represents an orthogonal transformation and its matrix elements relate to the combination of the Clebsch-Gordan coefficients. 

A transformation matrix that relates the uncoupled-spin and the coupled-spin kets can be generated by a recurrence procedure utilising only the Clebsch-Gordan coefficients. This can be applied to the uncoupled-spin interaction matrix in order to factorise the secular equation into blocks of lower dimension. 

After having the transformation propCTties of these operators, the composition of angnlar momentum operators can be applied using the rotation group Clebsch-Gordan coefficients... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

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