Levi-Civita symbol for vectors

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11: 34.10 Zeros

§34.10 Zeros

►In a $\mathit{3}j$ 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 $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ 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. …

12: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

13: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ 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 $\mathit{9}j$ 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). …

14: 1.1 Special Notation

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$x,y$	real variables.
…
$⟨f,g⟩$	inner, or scalar, product for real or complex vectors or functions.
…
$𝐮$, $𝐯$	column vectors.
${𝐄}_n$	the space of all $n$-dimensional vectors.
…

…

15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. …$V$ becomes a normed linear vector space. If $\Vert v\Vert =1$ then $v$ is normalized. Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. …

16: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

17: 21.1 Special Notation

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$g,h$	positive integers.
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$𝜶,𝜷$	$g$-dimensional vectors, with all elements in $[0,1)$, unless stated otherwise.
$a_j$	$j$th element of vector $𝐚$.
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$𝐚\cdot 𝐛$	scalar product of the vectors $𝐚$ and $𝐛$.
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$S^g$	set of $g$-dimensional vectors with elements in $S$.
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►Lowercase boldface letters or numbers are $g$-dimensional real or complex vectors, either row or column depending on the context. …

18: 21.6 Products

… ►that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${𝐓}^{\mathrm{T}}$ on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. …where ${𝐜}_j$ and ${𝐝}_j$ are arbitrary $h$-dimensional vectors. … ►Then …Thus $𝝂$ is a $g$-dimensional vector whose entries are either $0$ or $1$. …

19: 34.1 Special Notation

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$2j_1,2j_2,2j_3,2l_1,2l_2,2l_3$	nonnegative integers.
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►The main functions treated in this chapter are the Wigner $\mathit{3}j,\mathit{6}j,\mathit{9}j$ symbols, respectively, … ►An often used alternative to the $\mathit{3}j$ symbol is the Clebsch–Gordan coefficient ►

34.1.1

ⓘ

Symbols:

: $\mathit{3}j$ 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 $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

… ►For other notations for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

20: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

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