Clebsch-Gordan coefficient relations

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

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)). 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

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... 

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... 

The above procedure is therefore adequate to deal with all the ground states encountered in the LSMlMs) basis but can readily be adapted when it is desired to take account of spin-orbit effects by the use of the LSJMj) basis. Thus any given LSJMj) component is related to the LSMlMs) components of the parent LS state via the Clebsch-Gordan coefficients such that... 

The T-matrix element is expanded similarly. The reduced V- and T-matrix elements are obtained by inverting (7.36) using the orthonormality relations (3.71) of the spherical harmonics and (3.89) of the Clebsch-Gordan coefficients. 

The reduced Lippmann—Schwinger equations are obtained by expanding all the amplitudes of (6.73) according to (7.36) and again using the orthonormality relations (3.71,3.89) to eliminate the integral over k and the sum over Clebsch—Gordan coefficients in the expansion of the projection operator... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

Because the Clebsch-Gordan coefficient (jijUlj2M2 jM) vanishes unless the sum of the first two projection quantum numbers equals the third (jui + jU2 = ju), only one value of ki2 and only one value of ki2 survive on the right-hand-side of this last equation. Subsequently, we can cany out the sum over and ji2 and use the orthogonality relation of the Clebsch-Gordan coefficients (with respect to summing over jujcn and Ji2, ki2) to obtain the simplified expression, which we could have perhaps anticipated,... 

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. 

Employing the orthogonality relations of the Clebsch-Gordan coefficients and recalling eq. (9), one can show that the AD S-matrix elements... 

The 3-j symbols, in parentheses, are related to the Clebsch-Gordan coefficients. They are widely tabulated and easily computed for applications. 

For a given j, we may also consider the case with an orbital angular momentum quantum number V larger than j so that j = I — (thus, Z = Z + 1 if j shall adopt the same value as before). However, for j = 1 — and V > 0, we cannot exploit Eq. (4.142) directly, but need to rearrange the Clebsch-Gordan coefficient first so that the constraint for Eq. (4.142), J = ji + )2, is fulfilled. From well-known symmetry relations for Clebsch-Gordan coefficients [75] we find... 

Weissbluth s textbook provides an introduction to coupling of angular momenta in quantum mechanics and contains a survey of useful relations of Clebsch-Gordan coefficients as well as of Wigner 3/-symbols. This book can be considered elementary and a useful first step to the more involved presentations of the theory of angular momentum. 

Usine symmetry relations, Clebsch-Gordan coefficients... 

Finally, we emphasize that the Clebsch-Gordan coefficients for SU(2) are directly related to the Wigner s 3/ symbols where the latter have the advantage of being more symmetric than the former. 

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