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1: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …They therefore satisfy the triangle conditions …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by … When both conditions are satisfied the 3 j symbol can be expressed as the finite sum …
2: 34.11 Higher-Order 3 n j Symbols
§34.11 Higher-Order 3 n j Symbols
3: 34.10 Zeros
§34.10 Zeros
In a 3 j symbol, if the three angular momenta j 1 , j 2 , j 3 do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the 3 j symbol is zero. Similarly the 6 j symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four 3 j symbols in the summation. …However, the 3 j and 6 j symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …
4: 34.12 Physical Applications
§34.12 Physical Applications
The angular momentum coupling coefficients ( 3 j , 6 j , and 9 j symbols) are essential in the fields of nuclear, atomic, and molecular physics. … 3 j , 6 j , and 9 j symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).
5: 34.13 Methods of Computation
§34.13 Methods of Computation
Methods of computation for 3 j and 6 j symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). …
6: 34.14 Tables
§34.14 Tables
Tables of exact values of the squares of the 3 j and 6 j symbols in which all parameters are 8 are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of 3 j , 6 j , and 9 j symbols on pp. … Tables of 3 j and 6 j symbols in which all parameters are 17 / 2 are given in Appel (1968) to 6D. …Other tabulations for 3 j symbols are listed on pp. …
7: 34.1 Special Notation
2 j 1 , 2 j 2 , 2 j 3 , 2 l 1 , 2 l 2 , 2 l 3 nonnegative integers.
The main functions treated in this chapter are the Wigner 3 j , 6 j , 9 j symbols, respectively, … An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient
34.1.1 ( j 1 m 1 j 2 m 2 | j 1 j 2 j 3 m 3 ) = ( 1 ) j 1 j 2 + m 3 ( 2 j 3 + 1 ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ;
For other notations for 3 j , 6 j , 9 j symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
8: 34 3j, 6j, 9j Symbols
Chapter 34 3 j , 6 j , 9 j Symbols
9: 34.3 Basic Properties: 3 j Symbol
§34.3 Basic Properties: 3 j Symbol
Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to 3 j symbols, for which see Edmonds (1974, Chapter 4). …
10: 16.24 Physical Applications
§16.24(iii) 3 j , 6 j , and 9 j Symbols
The 3 j symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …