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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 …where r , s , t is any permutation of 1 , 2 , 3 . The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by …
2: 34.11 Higher-Order 3 n j Symbols
§34.11 Higher-Order 3 n j Symbols
3: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
π / 12 1 4 2 ( 3 1 ) 1 4 2 ( 3 + 1 ) 2 3 2 ( 3 + 1 ) 2 ( 3 1 ) 2 + 3
π / 6 1 2 1 2 3 1 3 3 2 2 3 3 3
π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
2 π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
5 π / 6 1 2 1 2 3 1 3 3 2 2 3 3 3
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: 34.3 Basic Properties: 3 j Symbol
§34.3 Basic Properties: 3 j Symbol
§34.3(ii) Symmetry
§34.3(iv) Orthogonality
§34.3(vi) Sums
6: 34.5 Basic Properties: 6 j Symbol
34.5.6 { j 1 j 2 j 3 1 j 3 1 j 2 + 1 } = ( 1 ) J ( ( J 2 j 2 1 ) ( J 2 j 2 ) ( J 2 j 3 + 1 ) ( J 2 j 3 + 2 ) ( 2 j 2 + 1 ) ( 2 j 2 + 2 ) ( 2 j 2 + 3 ) ( 2 j 3 1 ) 2 j 3 ( 2 j 3 + 1 ) ) 1 2 ,
34.5.9 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 1 1 2 ( j 2 + l 2 + j 3 l 3 ) 1 2 ( j 2 l 2 + j 3 + l 3 ) l 1 1 2 ( j 2 + l 2 j 3 + l 3 ) 1 2 ( j 2 + l 2 + j 3 + l 3 ) } ,
34.5.14 j 3 ( 2 j 3 + 1 ) ( 2 l 3 + 1 ) { j 1 j 2 j 3 l 1 l 2 l 3 } { j 1 j 2 j 3 l 1 l 2 l 3 } = δ l 3 , l 3 .
34.5.16 ( 1 ) j 1 + j 2 + j 3 + j 1 + j 2 + l 1 + l 2 { j 1 j 2 j 3 l 1 l 2 l 3 } { j 1 j 2 j 3 l 1 l 2 l 3 } = j ( 1 ) l 3 + l 3 + j ( 2 j + 1 ) { j 1 j 1 j j 2 j 2 j 3 } { l 3 l 3 j j 1 j 1 l 2 } { l 3 l 3 j j 2 j 2 l 1 } .
34.5.23 ( j 1 j 2 j 3 m 1 m 2 m 3 ) { j 1 j 2 j 3 l 1 l 2 l 3 } = m 1 m 2 m 3 ( 1 ) l 1 + l 2 + l 3 + m 1 + m 2 + m 3 ( j 1 l 2 l 3 m 1 m 2 m 3 ) ( l 1 j 2 l 3 m 1 m 2 m 3 ) ( l 1 l 2 j 3 m 1 m 2 m 3 ) .
7: 34.4 Definition: 6 j Symbol
§34.4 Definition: 6 j Symbol
The 6 j symbol is defined by the following double sum of products of 3 j symbols: …where the summation is taken over all admissible values of the m ’s and m ’s for each of the four 3 j symbols; compare (34.2.2) and (34.2.3). Except in degenerate cases the combination of the triangle inequalities for the four 3 j symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths j 1 , j 2 , j 3 , l 1 , l 2 , l 3 ; see Figure 34.4.1. … where F 3 4 is defined as in §16.2. …
8: 34.1 Special Notation
The main functions treated in this chapter are the Wigner 3 j , 6 j , 9 j symbols, respectively,
( j 1 j 2 j 3 m 1 m 2 m 3 ) ,
{ j 1 j 2 j 3 l 1 l 2 l 3 } ,
An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient …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).
9: 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. …
10: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
For large values of the parameters in the 3 j , 6 j , and 9 j symbols, different asymptotic forms are obtained depending on which parameters are large. …
34.8.1 { j 1 j 2 j 3 j 2 j 1 l 3 } = ( 1 ) j 1 + j 2 + j 3 + l 3 ( 4 π ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 l 3 + 1 ) sin θ ) 1 2 ( cos ( ( l 3 + 1 2 ) θ 1 4 π ) + o ( 1 ) ) , j 1 , j 2 , j 3 l 3 1 ,
Uniform approximations in terms of Airy functions for the 3 j and 6 j symbols are given in Schulten and Gordon (1975b). For approximations for the 3 j , 6 j , and 9 j symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.