# 3j symbols

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##### 1: 34.2 Definition: $\mathit{3j}$ Symbol
###### §34.2 Definition: $\mathit{3j}$Symbol
The quantities $j_{1},j_{2},j_{3}$ in the $\mathit{3j}$ 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 $\mathit{3j}$ symbol can be expressed as the finite sum …
##### 3: 34.10 Zeros
###### §34.10 Zeros
In a $\mathit{3j}$ 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 $\mathit{3j}$ symbol is zero. Similarly the $\mathit{6j}$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3j}$ symbols in the summation. …However, the $\mathit{3j}$ and $\mathit{6j}$ 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 ($\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ 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 $\mathit{3j}$ and $\mathit{6j}$ 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 $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3j},\mathit{6j}$, and $\mathit{9j}$ symbols on pp. … Tables of $\mathit{3j}$ and $\mathit{6j}$ symbols in which all parameters are $\leq 17/2$ are given in Appel (1968) to 6D. …Other tabulations for $\mathit{3j}$ symbols are listed on pp. …
##### 7: 34.1 Special Notation
 $2j_{1},2j_{2},2j_{3},2l_{1},2l_{2},2l_{3}$ nonnegative integers. …
The main functions treated in this chapter are the Wigner $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, respectively, … An often used alternative to the $\mathit{3j}$ symbol is the Clebsch–Gordan coefficient
34.1.1 $\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};$
For other notations for $\mathit{3j}$, $\mathit{6j}$, $\mathit{9j}$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
##### 9: 34.3 Basic Properties: $\mathit{3j}$ Symbol
###### §34.3 Basic Properties: $\mathit{3j}$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 $\mathit{3j}$ symbols, for which see Edmonds (1974, Chapter 4). …
##### 10: 16.24 Physical Applications
###### §16.24(iii) $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$Symbols
The $\mathit{3j}$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …