Clebsch-Gordan coefficient

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Clebsch-Gordan coefficient

[¦klepsh ¦gȯrd·ən ‚kō·ə′fish·ənt]
(quantum mechanics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Here [mathematical expression not reproducible] are the Clebsch-Gordan coefficients [23], where
Using (38)-(41), we obtain for Clebsch-Gordan coefficients the following properties:
McAllister, "On the computation of Clebsch-Gordan coefficients and the dilation effect," Experiment.
Here, [F.sub.i]([r.sub.1]), [F.sub.k]([r.sub.2]) denote radial functions of the appropriate electron, [j.sub.1], [j.sub.2], [[pi].sub.1], [[pi].sub.2] denote the single particle spin and parity of the electrons, respectively, J is the total spin obtained by using the appropriate Clebsch-Gordan coefficients [2,10] and M denotes the magnetic quantum number of the total angular momentum,
In each quark configuration, spin and spatial angular momentum are coupled to a total single particle j-value and the Clebsch-Gordan coefficients determine the portion of spin-up and spin-down of the quark.
A comparative analysis of the properties of these polynomials and [su.sub.q](2) and [su.sub.q](1, 1) Clebsch-Gordan coefficients shows that each relation for q-Hahn polynomials has the corresponding partner among the properties of q-CGC and vice versa.
Clebsch-Gordan coefficients, discrete orthogonal polynomials (q-discrete orthogonal polynomials), Nikiforov-Uvarov approach, quantum groups and algebras
Based on this well known fact we investigate the relation between the Clebsch-Gordan coefficients -also known as 3 j symbols[42] for the quantum algebras S[U.sub.q] (2) and S[U.sub.q] (1 ,1) with q-analogues of the Hahn polynomials on the non-uniform lattice x(s) = [q.sup.s] - 1 / q - 1.
Notice that (4.24) constitutes an extension--in this case a q-analog--of the well known relation between the classical Hahn polynomials and Clebsch-Gordan coefficients. Furthermore, this expression is in accordance with the results obtained in [33] (see also [1] for more details).
Now, the substitution of the parameters (4.22) into the expression (3.11) with the help of (4.24) gives the q-analog of the Racah formula for [su.sub.q] (2) Clebsch-Gordan coefficients Two particular cases are immediately derived from (4.25):
Relation between the Clebsch-Gordan coefficients for the quantum algebras [su.sub.q](2) and [su.sub.q](1, 1).
New problems about the Clebsch-Gordan coefficients have been raised recently by the specialists of computational complexity.