§ 34.3. Basic Properties: $3j$ Symbol

Permalink:: http://dlmf.nist.gov/34.3

§34.3(i)Special Cases
§34.3(ii)Symmetry
§34.3(iii)Recursion Relations
§34.3(iv)Orthogonality
§34.3(v)Generating Functions
§34.3(vi)Sums
§34.3(vii)Relations to Legendre Polynomials and Spherical Harmonics

§ 34.3(i). Special Cases

Notes:: See Edmonds (1974, pp. 48–50) and de-Shalit and Talmi (1963, p. 519).
Permalink:: http://dlmf.nist.gov/34.3.SS1

When any one of $j_{{1}},j_{{2}},j_{{3}}$ is equal to $0,\tfrac{1}{2}$ , or 1, the $3j$ symbol has a simple algebraic form. Examples are provided by

34.3.1 $\begin{pmatrix}j&j&0\\ m&-m&0\end{pmatrix}=\frac{(-1)^{{j-m}}}{(2j+1)^{{\frac{1}{2}}}},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E1
Encodings:: TeX, pMathML, png

34.3.2 $\begin{pmatrix}j&j&1\\ m&-m&0\end{pmatrix}=(-1)^{{j-m}}\frac{2m}{\left(2j(2j+1)(2j+2)\right)^{{\frac{1}{2}}}},$ $j\ge\tfrac{1}{2}$ ,

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E2
Encodings:: TeX, pMathML, png

34.3.3 $\begin{pmatrix}j&j&1\\ m&-m-1&1\end{pmatrix}=(-1)^{{j-m}}\left(\frac{2(j-m)(j+m+1)}{2j(2j+1)(2j+2)}\right)^{{\frac{1}{2}}},$ $j\ge\tfrac{1}{2}$ .

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E3
Encodings:: TeX, pMathML, png

For these and other results, and also cases in which any one of $j_{1},j_{2},j_{3}$ is $\frac{3}{2}$ or 2, see Edmonds (1974, pp. 125–127).

Next define

34.3.4 $J=j_{1}+j_{2}+j_{3}.$

Defines:: $J$ : sum
Symbols:: $j,j_{r}$ : nonnegative integer
Referenced by:: §34.5(i)
Permalink:: http://dlmf.nist.gov/34.3.E4
Encodings:: TeX, pMathML, png

Then assuming the triangle conditions are satisfied

34.3.5 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ 0&0&0\end{pmatrix}=\begin{cases}0,&\mbox{$J$ odd},\\ (-1)^{{\frac{1}{2}J}}\left(\dfrac{(J-2j_{1})!(J-2j_{2})!(J-2j_{3})!}{(J+1)!}\right)^{{\frac{1}{2}}}\dfrac{(\frac{1}{2}J)!}{(\frac{1}{2}J-j_{1})!(\frac{1}{2}J-j_{2})!(\frac{1}{2}J-j_{3})!},&\mbox{$J$ even}.\end{cases}$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.3.E5
Encodings:: TeX, TeX, TeX, pMathML, pMathML, pMathML, png, png, png

Lastly,

34.3.6 $\begin{pmatrix}j_{1}&j_{2}&j_{1}+j_{2}\\ m_{1}&m_{2}&-m_{1}-m_{2}\end{pmatrix}=(-1)^{{j_{{1}}-j_{{2}}+m_{{1}}+m_{{2}}}}\left(\frac{(2j_{{1}})!(2j_{{2}})!(j_{{1}}+j_{{2}}+m_{{1}}+m_{{2}})!(j_{{1}}+j_{{2}}-m_{{1}}-m_{{2}})!}{(2j_{{1}}+2j_{{2}}+1)!(j_{{1}}+m_{{1}})!(j_{{1}}-m_{{1}})!(j_{{2}}+m_{{2}})!(j_{{2}}-m_{{2}})!}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E6
Encodings:: TeX, pMathML, png

34.3.7 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ j_{1}&-j_{1}-m_{3}&m_{3}\end{pmatrix}=(-1)^{{-j_{2}+j_{3}+m_{3}}}\left(\frac{(2j_{1})!(-j_{1}+j_{2}+j_{3})!(j_{1}+j_{2}+m_{3})!(j_{3}-m_{3})!}{(j_{1}+j_{2}+j_{3}+1)!(j_{1}-j_{2}+j_{3})!(j_{1}+j_{2}-j_{3})!(-j_{1}+j_{2}-m_{3})!(j_{3}+m_{3})!}\right)^{{\frac{1}{2}}}.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(i)
Permalink:: http://dlmf.nist.gov/34.3.E7
Encodings:: TeX, pMathML, png

Again it is assumed that in (34.3.7) the triangle conditions are satisfied.

§ 34.3(ii). Symmetry

Notes:: See Edmonds (1974, pp. 46–47).
Permalink:: http://dlmf.nist.gov/34.3.SS2

Even permutations of columns of a $3j$ symbol leave it unchanged; odd permutations of columns produce a phase factor $(-1)^{{j_{{1}}+j_{{2}}+j_{{3}}}}$ , for example,

34.3.8 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{2}&j_{3}&j_{1}\\ m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}j_{3}&j_{1}&j_{2}\\ m_{3}&m_{1}&m_{2}\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(ii)
Permalink:: http://dlmf.nist.gov/34.3.E8
Encodings:: TeX, pMathML, png

34.3.9 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{{j_{{1}}+j_{{2}}+j_{{3}}}}\begin{pmatrix}j_{2}&j_{1}&j_{3}\\ m_{2}&m_{1}&m_{3}\end{pmatrix}.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E9
Encodings:: TeX, pMathML, png

Next,

34.3.10 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{{j_{{1}}+j_{{2}}+j_{{3}}}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ -m_{1}&-m_{2}&-m_{3}\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(ii)
Permalink:: http://dlmf.nist.gov/34.3.E10
Encodings:: TeX, pMathML, png

34.3.11 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}j_{1}&\frac{1}{2}(j_{{2}}+j_{{3}}+m_{{1}})&\frac{1}{2}(j_{{2}}+j_{{3}}-m_{{1}})\\ j_{{2}}-j_{{3}}&\frac{1}{2}(j_{{3}}-j_{{2}}+m_{{1}})+m_{{2}}&\frac{1}{2}(j_{{3}}-j_{{2}}+m_{{1}})+m_{{3}}\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(ii)
Permalink:: http://dlmf.nist.gov/34.3.E11
Encodings:: TeX, pMathML, png

34.3.12 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}(j_{{1}}+j_{{2}}-m_{{3}})&\frac{1}{2}(j_{{2}}+j_{{3}}-m_{{1}})&\frac{1}{2}(j_{{1}}+j_{{3}}-m_{{2}})\\ j_{{3}}-\frac{1}{2}(j_{{1}}+j_{{2}}+m_{{3}})&j_{{1}}-\frac{1}{2}(j_{{2}}+j_{{3}}+m_{{1}})&j_{{2}}-\frac{1}{2}(j_{{1}}+j_{{3}}+m_{{2}})\end{pmatrix}.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(ii)
Permalink:: http://dlmf.nist.gov/34.3.E12
Encodings:: TeX, pMathML, png

Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). See Srinivasa Rao and Rajeswari (1993, pp. 44–47) and references given there.

§ 34.3(iii). Recursion Relations

Notes:: See Edmonds (1974, p. 48).
Permalink:: http://dlmf.nist.gov/34.3.SS3

In the following three equations it is assumed that the triangle conditions are satisfied by each $3j$ symbol.

34.3.13 $\left((j_{{1}}+j_{{2}}+j_{{3}}+1)(-j_{{1}}+j_{{2}}+j_{{3}})\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{{2}}+m_{{2}})(j_{{3}}-m_{{3}})\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}&j_{{2}}-\frac{1}{2}&j_{{3}}-\frac{1}{2}\\ m_{{1}}&m_{{2}}-\frac{1}{2}&m_{{3}}+\frac{1}{2}\end{pmatrix}-\left((j_{{2}}-m_{{2}})(j_{{3}}+m_{{3}})\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}&j_{{2}}-\frac{1}{2}&j_{{3}}-\frac{1}{2}\\ m_{{1}}&m_{{2}}+\frac{1}{2}&m_{{3}}-\frac{1}{2}\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E13
Encodings:: TeX, pMathML, png

34.3.14 $\left(j_{{1}}(j_{{1}}+1)-j_{{2}}(j_{{2}}+1)-j_{{3}}(j_{{3}}+1)-2m_{{2}}m_{{3}}\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=\left((j_{{2}}-m_{{2}})(j_{{2}}+m_{{2}}+1)(j_{{3}}-m_{{3}}+1)(j_{{3}}+m_{{3}})\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ m_{{1}}&m_{{2}}+1&m_{{3}}-1\end{pmatrix}+\left((j_{{2}}-m_{{2}}+1)(j_{{2}}+m_{{2}})(j_{{3}}-m_{{3}})(j_{{3}}+m_{{3}}+1)\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ m_{{1}}&m_{{2}}-1&m_{{3}}+1\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E14
Encodings:: TeX, pMathML, png

34.3.15 $(2j_{{1}}+1)\left((j_{{2}}(j_{{2}}+1)-j_{{3}}(j_{{3}}+1))m_{{1}}-j_{{1}}(j_{{1}}+1)(m_{{3}}-m_{{2}})\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=(j_{{1}}+1)\left(j_{{1}}^{{2}}-(j_{{2}}-j_{{3}})^{{2}}\right)^{{\frac{1}{2}}}\left((j_{{2}}+j_{{3}}+1)^{{2}}-j_{{1}}^{{2}}\right)^{{\frac{1}{2}}}\left(j_{{1}}^{{2}}-m_{{1}}^{{2}}\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}-1&j_{{2}}&j_{{3}}\\ m_{{1}}&m_{{2}}&m_{{3}}\end{pmatrix}+j_{{1}}\left((j_{{1}}+1)^{{2}}-(j_{{2}}-j_{{3}})^{{2}}\right)^{{\frac{1}{2}}}\left((j_{{2}}+j_{{3}}+1)^{{2}}-(j_{{1}}+1)^{{2}}\right)^{{\frac{1}{2}}}\left((j_{{1}}+1)^{{2}}-m_{{1}}^{{2}}\right)^{{\frac{1}{2}}}\begin{pmatrix}j_{{1}}+1&j_{{2}}&j_{{3}}\\ m_{{1}}&m_{{2}}&m_{{3}}\end{pmatrix}.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E15
Encodings:: TeX, pMathML, png

For these and other recursion relations see Varshalovich et al. (1988, §8.6). See also Micu (1968), Louck (1958), Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220–225), and Luscombe and Luban (1998).

§ 34.3(iv). Orthogonality

Notes:: See Edmonds (1974, p. 47), de-Shalit and Talmi (1963, p. 515).
Permalink:: http://dlmf.nist.gov/34.3.SS4

34.3.16 $\sum _{{m_{1}m_{2}}}(2j_{3}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j^{{\prime}}_{3}\\ m_{1}&m_{2}&m^{{\prime}}_{3}\end{pmatrix}=\delta _{{j_{{3}},j^{{\prime}}_{{3}}}}\delta _{{m_{{3}},m^{{\prime}}_{{3}}}},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(iv), §34.9
Permalink:: http://dlmf.nist.gov/34.3.E16
Encodings:: TeX, pMathML, png

34.3.17 $\sum _{{j_{{3}}m_{{3}}}}(2j_{{3}}+1)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m^{{\prime}}_{1}&m^{{\prime}}_{2}&m_{3}\end{pmatrix}=\delta _{{m_{{1}},m^{{\prime}}_{{1}}}}\delta _{{m_{{2}},m^{{\prime}}_{{2}}}},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E17
Encodings:: TeX, pMathML, png

34.3.18 $\sum _{{m_{{1}}m_{{2}}m_{{3}}}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}=1.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.3(iv)
Permalink:: http://dlmf.nist.gov/34.3.E18
Encodings:: TeX, pMathML, png

In the summations (34.3.16) – (34.3.18) the summation variables range over all values that satisfy the conditions given in (34.2.1) – (34.2.3). Similar conventions apply to all subsequent summations in this chapter.

§ 34.3(v). Generating Functions

Permalink:: http://dlmf.nist.gov/34.3.SS5

For generating functions for the $3j$ symbol see Biedenharn and van Dam (1965, p. 245, Eq. (3.42) and p. 247, Eq. (3.55)).

§ 34.3(vi). Sums

Permalink:: http://dlmf.nist.gov/34.3.SS6

For sums of products of $3j$ symbols, see Varshalovich et al. (1988, pp. 259–262).

§ 34.3(vii). Relations to Legendre Polynomials and Spherical Harmonics

Notes:: See Thompson (1994, p. 288), Edmonds (1974, p. 63).
Permalink:: http://dlmf.nist.gov/34.3.SS7

For the polynomials $P_{{l}}$ see §Ch.18, and for the functions $Y_{{{l},{m}}}$ and ${Y_{{{l},{m}}}^{{*}}}$ see §Ch.14.

34.3.19 $P_{{l_{1}}}\!\left(\cos\theta\right)P_{{l_{2}}}\!\left(\cos\theta\right)=\sum _{l}(2l+1)\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}P_{{l}}\!\left(\cos\theta\right),$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E19
Encodings:: TeX, pMathML, png

34.3.20 $Y_{{{l_{1}},{m_{1}}}}\!\left(\theta,\phi\right)Y_{{{l_{2}},{m_{2}}}}\!\left(\theta,\phi\right)=\sum _{{l,m}}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{{\frac{1}{2}}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}{Y_{{{l},{m}}}^{{*}}}\!\left(\theta,\phi\right)\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E20
Encodings:: TeX, pMathML, png

34.3.21 $\int _{0}^{\pi}P_{{l_{1}}}\!\left(\cos\theta\right)P_{{l_{2}}}\!\left(\cos\theta\right)P_{{l_{3}}}\!\left(\cos\theta\right)\sin\theta d\theta=2\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}^{2},$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.3.E21
Encodings:: TeX, pMathML, png

34.3.22 $\int _{0}^{{2\pi}}\!\int _{0}^{{\pi}}Y_{{{l_{1}},{m_{1}}}}\!\left(\theta,\phi\right)Y_{{{l_{2}},{m_{2}}}}\!\left(\theta,\phi\right)Y_{{{l_{3}},{m_{3}}}}\!\left(\theta,\phi\right)\sin\theta d\theta d\phi=\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}\right)^{{\frac{1}{2}}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.$

Symbols:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E22
Encodings:: TeX, pMathML, png

Equations (34.3.19) – (34.3.22) are particular cases of more general results that relate rotation matrices to $3j$ symbols, for which see Edmonds (1974, Chapter 4). The left- and right-hand sides of (34.3.22) are known, respectively, as Gaunt's integral and the Gaunt coefficient (Gaunt (1929)).

§ 34.3. Basic Properties: Symbol

Contents