Clebsch-Gordan

Clebsch-Gordan coefficients coupling ground and excited levels that = transitions coupled by linearly... 

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. 

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

The conditions determined by the selection rules for Clebsch-Gordan coefficients and 6j symbols provide the following block-diagonal structure of the operator... 

The asymptotic expressions for the corresponding Clebsch-Gordan coefficients take the form... 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

Now the pairs of rotation matrix products in Eq. (A.7) can be replaced with Clebsch-Gordan series... 

By reexpressing the spherical harmonics in the form of D rotation matrices they may be effectively substituted by a Clebsch-Gordan series yielding... 

M-sum unitarity of the first two Clebsch-Gordan coefficients now means that the sum over p reduces to a simple delta function ... 

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

Where appropriate the spin components are then coupled via the Clebsch-Gordan coefficients. In this way therefore the wave functions for a given dx configuration may be obtained from the dx 1 results. 

The products of second-rank tensor components, such as A1-, j ( P/ JA1-, j (0Fi), can be expressed in coupled form using the Clebsch-Gordan coefficients, yielding tensor terms, of spatial rank / = 0, 2, and 4 ... 

The expansion coefficients on the right hand side of Eq. (1.25) are the Clebsch-Gordan coefficients.2 The eigenfunctions of the angular momentum, which can be written, abstractly, using Dirac notation 11, m >, satisfy the equations (h=1)... 

Inserting the appropriate values of the Clebsch-Gordan coefficients, one obtains... 

The result (2.167) is particularly important, since it is used to analyze experimental data. It is merely a consequence of the fact that the quadrupole operator is a tensor of rank 2. Sj is just the square of the Clebsch-Gordan coefficients in... 

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two 0(4) and two 0(3) representations, respectively. They can be evaluated either analytically using Racah s factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2... 

We define the Clebsch-Gordan coefficients with the usual (Condon and Short-ley, 1967) phase convention... 

The Clebsch-Gordan coefficients satisfy the orthogonality relations... 

Instead of Clebsch-Gordan coefficients it is often convenient to use the Wigner 3 — j symbols... 

The Clebsch-Gordan series for S0(4) can then be simply constructed and is given by... 

Sharp, R. T. (1960), Simple Derivation of the Clebsch-Gordan Coefficients, Am. J. Phys. 28, 116. 

We can pass from tree a to b using the suitable Clebsch-Gordan coeficient (eq. 12). The tree (c) illustrates the hyperspherical parametrization that leads to the hyperspherical harmonics Yn- Xm(, W 9) They are related to the harmonics of tree a through the Z coeficient defined in eg. (15). The connection between (b) and (c) requires a Clebsch Gordan coefficient and a phase change related to a (see eq. (14)). 

---