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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: 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 / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
4.17.1 lim z 0 sin z z = 1 ,
4.17.2 lim z 0 tan z z = 1 .
4.17.3 lim z 0 1 - cos z z 2 = 1 2 .
3: 34.3 Basic Properties: 3 j Symbol
§34.3 Basic Properties: 3 j Symbol
When any one of j 1 , j 2 , j 3 is equal to 0 , 1 2 , or 1 , the 3 j symbol has a simple algebraic form. …For these and other results, and also cases in which any one of j 1 , j 2 , j 3 is 3 2 or 2 , see Edmonds (1974, pp. 125–127). …
§34.3(ii) Symmetry
Even permutations of columns of a 3 j symbol leave it unchanged; odd permutations of columns produce a phase factor ( - 1 ) j 1 + j 2 + j 3 , for example, …
4: 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. …
5: 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 ) ,
An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient …see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). 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).
6: 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. … and the symbol o ( 1 ) denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … 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.
7: 34.5 Basic Properties: 6 j Symbol
If any lower argument in a 6 j symbol is 0 , 1 2 , or 1 , then the 6 j symbol has a simple algebraic form. …
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 } .
8: 22.9 Cyclic Identities
Throughout this subsection m and p are positive integers with 1 m p . … In this subsection 1 m p and 1 n p . … These identities are cyclic in the sense that each of the indices m , n in the first product of, for example, the form s m , p ( 4 ) s n , p ( 4 ) are simultaneously permuted in the cyclic order: m m + 1 m + 2 p 1 2 m - 1 ; n n + 1 n + 2 p 1 2 n - 1 . …
§22.9(iii) Typical Identities of Rank 3
22.9.23 s 1 , 3 ( 4 ) d 1 , 3 ( 4 ) c 2 , 3 ( 4 ) c 3 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) c 3 , 3 ( 4 ) c 1 , 3 ( 4 ) + s 3 , 3 ( 4 ) d 3 , 3 ( 4 ) c 1 , 3 ( 4 ) c 2 , 3 ( 4 ) = κ 2 1 - κ 2 ( s 1 , 3 ( 4 ) d 1 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) + s 2 , 3 ( 4 ) d 2 , 3 ( 4 ) ) .
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: 4.43 Cubic Equations
A = ( - 4 3 p ) 1 / 2 ,
B = ( 4 3 p ) 1 / 2 .
  • (a)

    A sin a , A sin ( a + 2 3 π ) , and A sin ( a + 4 3 π ) , with sin ( 3 a ) = 4 q / A 3 , when 4 p 3 + 27 q 2 0 .

  • (b)

    A cosh a , A cosh ( a + 2 3 π i ) , and A cosh ( a + 4 3 π i ) , with cosh ( 3 a ) = - 4 q / A 3 , when p < 0 , q < 0 , and 4 p 3 + 27 q 2 > 0 .

  • (c)

    B sinh a , B sinh ( a + 2 3 π i ) , and B sinh ( a + 4 3 π i ) , with sinh ( 3 a ) = - 4 q / B 3 , when p > 0 .