Levi-Civita symbol

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1: 34.6 Definition: $\mathit{9j}$ Symbol

§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

►

34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

2: 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: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

►The

\mathit{6j}

symbol is defined by the following double sum of products of

\mathit{3j}

symbols: …where the summation is taken over all admissible values of the

m

’s and

m^{\prime}

’s for each of the four

\mathit{3j}

symbols; compare (34.2.2) and (34.2.3). … ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►where

{{}_{4}F_{3}}

is defined as in §16.2. …

4: 1.6 Vectors and Vector-Valued Functions

… ►

Levi-Civita Symbol

►

1.6.14

\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even % permutation of }1,2,3,\\ -1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\ \phantom{-}0,&\text{otherwise}.\end{cases}

ⓘ

Defines:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol
Symbols:: $j$ : integer and $k$ : integer
Permalink:: http://dlmf.nist.gov/1.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

1.6.16

\epsilon_{jk\ell}\epsilon_{\ell mn}=\delta_{j,m}\delta_{k,n}-\delta_{j,n}% \delta_{k,m},

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol, $j$ : integer, $k$ : integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.6.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

1.6.17

\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk\ell}\mathbf{e}_{\ell};

ⓘ

Symbols:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol, $j$ : integer, $k$ : integer and $\mathbf{e}_{j}$ : unit vectors
Permalink:: http://dlmf.nist.gov/1.6.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

… ►

1.6.18

a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=\epsilon_{jk\ell}a_{j}b_{k}% \mathbf{e}_{\ell};

ⓘ

Symbols:: $\epsilon_{\NVar{j}\NVar{k}\NVar{\ell}}$ : Levi-Civita symbol, $j$ : integer, $k$ : integer and $\mathbf{e}_{j}$ : unit vectors
Permalink:: http://dlmf.nist.gov/1.6.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §1.6(ii), §1.6(ii), §1.6 and Ch.1

…

5: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

►For information on

12j,15j

,…, symbols, see Varshalovich et al. (1988, §10.12) and Yutsis et al. (1962, pp. 62–65 and 122–153).

8: 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. Such zeros are called nontrivial zeros. …

9: 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).

10: 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). ►For

\mathit{9j}

symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

Levi-Civita symbol

1—10 of 210 matching pages

1: 34.6 Definition: $\mathit{9j}$ Symbol

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

2: 34.2 Definition: $\mathit{3j}$ Symbol

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

3: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

4: 1.6 Vectors and Vector-Valued Functions

Levi-Civita Symbol

5: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols

6: 27.9 Quadratic Characters

§27.9 Quadratic Characters

7: 34.14 Tables

§34.14 Tables

8: 34.10 Zeros

§34.10 Zeros

9: 34.12 Physical Applications

§34.12 Physical Applications

10: 34.13 Methods of Computation

§34.13 Methods of Computation

Levi-Civita symbol

1—10 of 210 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

2: 34.2 Definition: 3 ⁢ j Symbol

§34.2 Definition: 3 ⁢ j Symbol

3: 34.4 Definition: 6 ⁢ j Symbol

§34.4 Definition: 6 ⁢ j Symbol

4: 1.6 Vectors and Vector-Valued Functions

Levi-Civita Symbol

5: 34.11 Higher-Order 3 ⁢ n ⁢ j Symbols

§34.11 Higher-Order 3 ⁢ n ⁢ j Symbols

6: 27.9 Quadratic Characters

§27.9 Quadratic Characters

7: 34.14 Tables

§34.14 Tables

8: 34.10 Zeros

§34.10 Zeros

9: 34.12 Physical Applications

§34.12 Physical Applications

10: 34.13 Methods of Computation

§34.13 Methods of Computation

1: 34.6 Definition: $\mathit{9j}$ Symbol

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

2: 34.2 Definition: $\mathit{3j}$ Symbol

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

3: 34.4 Definition: $\mathit{6j}$ Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

5: 34.11 Higher-Order $3nj$ Symbols

§34.11 Higher-Order $3nj$ Symbols