3-j symbol

In quantum mechanics, the Wigner 3-j symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta.[1] While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically.

Mathematical relation to Clebsch–Gordan coefficients

The 3-j symbols are given in terms of the Clebsch–Gordan coefficients by

{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,(-m_{3})\rangle .

The j and m components are angular-momentum quantum numbers, i.e., every $j$ (and every corresponding $m$ ) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution $m 3 \to - m 3$ :

\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle =(-1)^{-j_{1}+j_{2}-m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.

Definitional relation to Clebsch–Gordan coefficients

The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third:

|j_{3}\,m_{3}\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle .

The 3-j symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero:

\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\sum _{m_{3}=-j_{3}}^{j_{3}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{3}m_{3}\rangle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=|0\,0\rangle .

Here $|0\,0\rangle$ is the zero-angular-momentum state ( $j=m=0$ ). It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient.

Since the state $|0\,0\rangle$ is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-j symbol is invariant under rotations.

Selection rules

The Wigner 3-j symbol is zero unless all these conditions are satisfied:

{\begin{aligned}&m_{i}\in \{-j_{i},-j_{i}+1,-j_{i}+2,\ldots ,j_{i}\}\quad (i=1,2,3),\\&m_{1}+m_{2}+m_{3}=0,\\&|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2},\\&(j_{1}+j_{2}+j_{3}){\text{ is an integer (and, moreover, an even integer if }}m_{1}=m_{2}=m_{3}=0{\text{)}}.\\\end{aligned}}

Symmetry properties

A 3-j symbol is invariant under an even permutation of its columns:

{\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}}.

An odd permutation of the columns gives a phase factor:

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

Changing the sign of the $m$ quantum numbers (time reversal) also gives a phase:

{\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}}.

The 3-j symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.[2] These symmetries are:

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

{\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}{\frac {j_{2}+j_{3}+m_{1}}{2}}&{\frac {j_{1}+j_{3}+m_{2}}{2}}&{\frac {j_{1}+j_{2}+m_{3}}{2}}\\j_{1}-{\frac {j_{2}+j_{3}-m_{1}}{2}}&j_{2}-{\frac {j_{1}+j_{3}-m_{2}}{2}}&j_{3}-{\frac {j_{1}+j_{2}-m_{3}}{2}}\end{pmatrix}}.

With the Regge symmetries, the 3-j symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-j symbol and assumes the properties of a semi-magic square:[3]

R={\begin{array}{|ccc|}\hline -j_{1}+j_{2}+j_{3}&j_{1}-j_{2}+j_{3}&j_{1}+j_{2}-j_{3}\\j_{1}-m_{1}&j_{2}-m_{2}&j_{3}-m_{3}\\j_{1}+m_{1}&j_{2}+m_{2}&j_{3}+m_{3}\\\hline \end{array}},

whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.[4]

Orthogonality relations

A system of two angular momenta with magnitudes $j 1$ and $j 2$ can be described either in terms of the uncoupled basis states (labeled by the quantum numbers $m 1$ and $m 2$ ), or the coupled basis states (labeled by $j 3$ and $m 3$ ). The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations

(2j_{3}+1)\sum _{m_{1}m_{2}}{\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}}=\delta _{j_{3},j'_{3}}\delta _{m_{3},m'_{3}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}},

\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_{1}'&m_{2}'&m_{3}\end{pmatrix}}=\delta _{m_{1},m_{1}'}\delta _{m_{2},m_{2}'}.

The triangular delta ${j 1 j 2 j 3}$ is equal to 1 when the triad (j₁, j₂, j₃) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called[5] a "3-j symbol" (without the m) in analogy to 6-j and 9-j symbols, all of which are irreducible summations of 3-jm symbols where no $m$ variables remain.

Relation to spherical harmonics

The 3-jm symbols give the integral of the products of three spherical harmonics

{\begin{aligned}&\int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&\quad ={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\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}}\end{aligned}}

with $l_{1}$ , $l_{2}$ and $l_{3}$ integers.

Relation to integrals of spin-weighted spherical harmonics

Similar relations exist for the spin-weighted spherical harmonics if $s_{1}+s_{2}+s_{3}=0$ :

{\begin{aligned}&\int d\mathbf {\hat {n}} \,_{s_{1}}\!Y_{j_{1}m_{1}}(\mathbf {\hat {n}} )\,_{s_{2}}\!Y_{j_{2}m_{2}}(\mathbf {\hat {n}} )\,_{s_{3}}\!Y_{j_{3}m_{3}}(\mathbf {\hat {n}} )\\&\quad ={\sqrt {\frac {(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)}{4\pi }}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}.\end{aligned}}

Recursion relations

{\begin{aligned}&{-}{\sqrt {(l_{3}\mp s_{3})(l_{3}\pm s_{3}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}&s_{3}\pm 1\end{pmatrix}}=\\&\quad ={\sqrt {(l_{1}\mp s_{1})(l_{1}\pm s_{1}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}\pm 1&s_{2}&s_{3}\end{pmatrix}}+{\sqrt {(l_{2}\mp s_{2})(l_{2}\pm s_{2}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}\pm 1&s_{3}\end{pmatrix}}.\end{aligned}}

Asymptotic expressions

For $l_{1}\ll l_{2},l_{3}$ a non-zero 3-j symbol is

{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}},

where $\cos(\theta )=-2m_{3}/(2l_{3}+1)$ , and $d_{mn}^{l}$ is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}},

where $\cos(\theta )=(m_{2}-m_{3})/(l_{2}+l_{3}+1)$ .

Metric tensor

The following quantity acts as a metric tensor in angular-momentum theory and is also known as a Wigner 1-jm symbol:[1]

{\begin{pmatrix}j\\m\quad m'\end{pmatrix}}:={\sqrt {2j+1}}{\begin{pmatrix}j&0&j\\m&0&m'\end{pmatrix}}=(-1)^{j-m'}\delta _{m,-m'}.

It can be used to perform time reversal on angular momenta.

Other properties

\sum _{m}(-1)^{j-m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {2j+1}}\,\delta _{J,0}.

{\frac {1}{2}}\int _{-1}^{1}P_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)\,dx={\begin{pmatrix}l&l_{1}&l_{2}\\0&0&0\end{pmatrix}}^{2},

where P are Legendre polynomials.

Relation to Racah $V$ -coefficients

Wigner 3-j symbols are related to Racah $V$ -coefficients[6] by a simple phase:

V(j_{1}\,j_{2}\,j_{3};m_{1}\,m_{2}\,m_{3})=(-1)^{j_{1}-j_{2}-j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}.

References

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Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento. 10 (3): 544. Bibcode:1958NCim...10..544R. doi:10.1007/BF02859841. S2CID 122299161.
Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
P. E. S. Wormer; J. Paldus (2006). "Angular Momentum Diagrams". Advances in Quantum Chemistry. Elsevier. 51: 59–124. Bibcode:2006AdQC...51...59W. doi:10.1016/S0065-3276(06)51002-0. ISBN 9780120348510. ISSN 0065-3276.
Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438.

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Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

External links

Stone, Anthony. "Wigner coefficient calculator".
Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". Archived from the original on 2007-09-29. (Numerical)
Stevenson, Paul (2002). "Clebsch-O-Matic". Computer Physics Communications. 147 (3): 853–858. Bibcode:2002CoPhC.147..853S. doi:10.1016/S0010-4655(02)00462-9.
369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)
Frederik J Simons: Matlab software archive, the code THREEJ.M
Sage (mathematics software) Gives exact answer for any value of j, m
Johansson, H. T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
Johansson, H. T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)

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[Wigner1951-1] Wigner, E. P. (1951). "On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups". In Wightman, Arthur S. (ed.). The Collected Works of Eugene Paul Wigner. 3. pp. 608–654. doi:10.1007/978-3-662-02781-3_42. ISBN 978-3-642-08154-5.

[2] Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento. 10 (3): 544. Bibcode:1958NCim...10..544R. doi:10.1007/BF02859841. S2CID 122299161.

[3] Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

[4] Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

[WormerPaldus2006-5] P. E. S. Wormer; J. Paldus (2006). "Angular Momentum Diagrams". Advances in Quantum Chemistry. Elsevier. 51: 59–124. Bibcode:2006AdQC...51...59W. doi:10.1016/S0065-3276(06)51002-0. ISBN 9780120348510. ISSN 0065-3276.

[6] Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438.