threej symbols

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1—10 of 18 matching pages

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 …

2: 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. …

3: 34.3 Basic Properties: $\mathit{3j}$ Symbol

§34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►

§34.3(ii) Symmetry

… ►

§34.3(iv) Orthogonality

… ►

§34.3(vi) Sums

… ►

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: 34.9 Graphical Method

§34.9 Graphical Method

… ►For specific examples of the graphical method of representing sums involving the

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

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};

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

… ►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).

8: 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).

9: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

…

10: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►Uniform approximations in terms of Airy functions for the

\mathit{3j}

and

\mathit{6j}

symbols are given in Schulten and Gordon (1975b). For approximations for the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

threej symbols

1—10 of 18 matching pages

1: 34.2 Definition: $\mathit{3j}$ Symbol

§34.2 Definition: $\mathit{3j}$ Symbol

2: 34.10 Zeros

§34.10 Zeros

3: 34.3 Basic Properties: $\mathit{3j}$ Symbol

§34.3 Basic Properties: $\mathit{3j}$ Symbol

§34.3(ii) Symmetry

§34.3(iv) Orthogonality

§34.3(vi) Sums

4: 34.14 Tables

§34.14 Tables

5: 34.13 Methods of Computation

§34.13 Methods of Computation

6: 34.9 Graphical Method

§34.9 Graphical Method

7: 34.1 Special Notation

8: 34.12 Physical Applications

§34.12 Physical Applications

9: 34.6 Definition: $\mathit{9j}$ Symbol

10: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

threej symbols

1—10 of 18 matching pages

1: 34.2 Definition: 3 ⁢ j Symbol

§34.2 Definition: 3 ⁢ j Symbol

2: 34.10 Zeros

§34.10 Zeros

3: 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

4: 34.14 Tables

§34.14 Tables

5: 34.13 Methods of Computation

§34.13 Methods of Computation

6: 34.9 Graphical Method

§34.9 Graphical Method

7: 34.1 Special Notation

8: 34.12 Physical Applications

§34.12 Physical Applications

9: 34.6 Definition: 9 ⁢ j Symbol

10: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

1: 34.2 Definition: $\mathit{3j}$ Symbol

§34.2 Definition: $\mathit{3j}$ Symbol

3: 34.3 Basic Properties: $\mathit{3j}$ Symbol

§34.3 Basic Properties: $\mathit{3j}$ Symbol

9: 34.6 Definition: $\mathit{9j}$ Symbol