Coefficient Wigner

The coefficients 0 are variously called angular momentum addition coefficients, or Wigner coefficients, or Clebsch-Gordan coefficients. Their importance for quantum mechanics was" first recognized by Wigner,6 who also provided a formula and a complete theory of them. The notation varies among different authors who deal with them7 ours follows most closely that of Rose. 

The Clebsch-Gordan coefficient may be expressed in terms of the Wigner coefficient (symbol in round brackets) ... 

As we can see from Eq. (6.8), the Wigner coefficients are much more symmetrical than the Clebsch-Gordan coefficients. Therefore, usually their tables are presented (see, for example, in [11] the tables for 71+72+73 < 16). 

The sign + or — has the same meaning as in the case of the Wigner coefficient (Fig. 8.1). Thus, graphs 8.4a and 8.46 correspond to the same Clebsch-Gordan coefficient... 

Its graphical representation is given in Fig. 8.5a. Therefore, unlike the case of the Wigner coefficients, for the Clebsch-Gordan coefficients there is no necessity to indicate the arrows on the lines. 

The righthand side contains the Wigner coefficient SjmTmi<7> o = -1, 0, +1, which expresses the usual rotation group selection rules... 

If this condition is not fulfilled, the Clebsch-Gordan coefficients vanish, otherwise they have certain numerical values (see Table 7.1). The Clebsch-Gordan coefficients are related to the Wigner coefficients [Wig51], also called 3j symbols jt = a, h = b, j3 = c,m1 = a, m2 = P,m3 = y), defined by... 

Assume that all elements of P3 leave the Hamiltonian 3 invariant. Then we can classify the states of the total system both by their S values and by the irreducible representations of P3. To determine which these latter representations are, we need to know the characters of the transpositions and of the cycles of order three for each pair of Vcdues (Si, S). Consider first the element (12). Because of the behaviour of the Wigner coefficients under interchange of angular momenta (see e.g. Ref. 20), p. 24) we have... 

When induction operators of high-order multipoles are taken into account intensity calculations tend to become very cumbersome [30,31]. We propose a relatively easy way of performing these calculations using the irreducible spherical tensor theory of multipole light scattering [e.g., Eqs. (6) and (7)] together with symbolic calculations of the Wigner coefficients by computer. 

Here we have used the complex conjugate of Eq. (11) and the fact that the Wigner coefficients are real quantities in both representations. Then, by comparison with the right-hand side of Eq. (27) the identity of Eq. (29) follows. 

The factor involving the Wigner coefficients may also be put into a simpler form it is invariant against rotation of the axis of quantization and must therefore be expressible in terms of invariants formed from the total spin operators of the two subsystems. On denoting the numerical factor in (37) by /i(5x, Sb, 5), it follows easily [16] that... 

Calculated from Clebsch-Gordon- and Wigner-coefficients of Ref. 15... 

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