DLMF: Untitled Document

… ►

34.5.6

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j% _{3}+1)(J-2j_{3}+2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}% \right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

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34.5.9

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_% {3}-l_{3})&\frac{1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(ii), §34.5 and Ch.34

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34.5.14

\sum_{j_{3}}(2j_{3}+1)(2l_{3}+1)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l^{\prime}_{3}\end{Bmatrix}=\delta_{l_{3},l^{\prime}_{3}}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.5.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(iv), §34.5 and Ch.34

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34.5.16

(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

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34.5.23

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m^{\prime}_{1}m^{\prime}_{2}m^{\prime}_{3% }}(-1)^{l_{1}+l_{2}+l_{3}+m^{\prime}_{1}+m^{\prime}_{2}+m^{\prime}_{3}}\begin{% pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(vi), §34.5 and Ch.34

…

§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). ►Except in degenerate cases the combination of the triangle inequalities for the four

\mathit{3j}

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

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

is defined as in §16.2. …

… ►The main functions treated in this chapter are the Wigner

\mathit{3j},\mathit{6j},\mathit{9j}

symbols, respectively, ►

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},

►

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},

… ►An often used alternative to the

\mathit{3j}

symbol is the Clebsch–Gordan coefficient …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).

§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. …

§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. … ►

34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

… ►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.

… ►

22.9.10

d_{1,3}^{(2)}d_{2,3}^{(2)}+d_{2,3}^{(2)}d_{3,3}^{(2)}+d_{3,3}^{(2)}d_{1,3}^{(2% )}=d_{1,3}^{(4)}d_{2,3}^{(4)}+d_{2,3}^{(4)}d_{3,3}^{(4)}+d_{3,3}^{(4)}d_{1,3}^% {(4)}=\kappa(\kappa+2).

ⓘ

Symbols:: $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Referenced by:: §22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)
Permalink:: http://dlmf.nist.gov/22.9.E10
Encodings:: pMML, png, TeX
Correction (effective with 1.0.25):: Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$ . It has been corrected by writing $p=3$ throughout.
Suggested 2019-11-07 by Juan Miguel Nieto
See also:: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

… ►

§22.9(iii) Typical Identities of Rank 3

… ►

22.9.13

s_{1,3}^{(4)}s_{2,3}^{(4)}s_{3,3}^{(4)}=-\frac{1}{1-\kappa^{2}}\left(s_{1,3}^{% (4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),

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Symbols:: $s_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.22

s_{1,3}^{(2)}c_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}+s_{2,3}^{(2)}c_{2,3}^{(2)% }d_{3,3}^{(2)}d_{1,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}d_{1,3}^{(2)}d_{2,3}^{(2% )}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}}\left(s_{1,3}^{(2)}c_{1,3}^{(2)}+s_{% 2,3}^{(2)}c_{2,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}\right),

ⓘ

Symbols:: $k$ : modulus, $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

►

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% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).

ⓘ

Symbols:: $s_{m,p}^{(r)}$ : cyclic quantity, $c_{m,p}^{(r)}$ : cyclic quantity, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

Chapter 3 Numerical Methods

…

… ►

A=\left(-\tfrac{4}{3}p\right)^{1/2},

►

B=\left(\tfrac{4}{3}p\right)^{1/2}.

… ►

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=4q/A^{3}$ , when $4p^{3}+27q^{2}\leq 0$ .

►

(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=-4q/A^{3}$ , when $p<0$ , $q<0$ , and $4p^{3}+27q^{2}>0$ .

►

(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

…

… ►

See accompanying text — Figure 7.3.1: Complementary error functions $\operatorname{erfc}x$ and $\operatorname{erfc}\left(10x\right)$ , $-3\leq x\leq 3$ . Magnify

… ►

►

… ►Then

\Delta>0

and the parallelogram with vertices at

0

,

2\omega_{1}

,

2\omega_{1}+2\omega_{3}

,

2\omega_{3}

is a rectangle. … ►Also,

e_{2}

and

g_{3}

have opposite signs unless

\omega_{3}=i\omega_{1}

, in which event both are zero. ►As functions of

\Im\omega_{3}

,

e_{1}

and

e_{2}

are decreasing and

e_{3}

is increasing. … ►The parallelogram

0

,

2\omega_{1}

,

2\omega_{1}+2\omega_{3}

,

2\omega_{3}

is a square, and … ►The parallelogram

0

,

2\omega_{1}-2\omega_{3}

,

2\omega_{1}

,

2\omega_{3}

, is a rhombus: see Figure 23.5.1. …

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21—30 of 569 matching pages

21: 34.5 Basic Properties: $\mathit{6j}$ Symbol

22: 34.4 Definition: $\mathit{6j}$ Symbol

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

23: 34.1 Special Notation

24: 34.10 Zeros

§34.10 Zeros

25: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

26: 22.9 Cyclic Identities

§22.9(iii) Typical Identities of Rank 3

27: 3 Numerical Methods

Chapter 3 Numerical Methods

28: 4.43 Cubic Equations

29: 7.3 Graphics

30: 23.5 Special Lattices

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21—30 of 569 matching pages

21: 34.5 Basic Properties: 6 ⁢ j Symbol

22: 34.4 Definition: 6 ⁢ j Symbol

§34.4 Definition: 6 ⁢ j Symbol

23: 34.1 Special Notation

24: 34.10 Zeros

§34.10 Zeros

25: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

26: 22.9 Cyclic Identities

§22.9(iii) Typical Identities of Rank 3

27: 3 Numerical Methods

Chapter 3 Numerical Methods

28: 4.43 Cubic Equations

29: 7.3 Graphics

30: 23.5 Special Lattices

21: 34.5 Basic Properties: $\mathit{6j}$ Symbol

22: 34.4 Definition: $\mathit{6j}$ Symbol

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