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In quantum mechanics, the Wigner 3-j symbols, also called 3j or 3-jm symbols, are related to Clebsch–Gordan coefficients through

\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle. \)


Inverse relation

The inverse relation can be found by noting that j1 − j2 − m3 is an integer and making the substitution \(m_3 \rightarrow -m_3 \) :

\( \langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{-j_1+j_2-m_3}\sqrt{2j_3+1} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix}. \)

Symmetry properties

The symmetry properties of 3j symbols are more convenient than those of Clebsch–Gordan coefficients. A 3j 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}. \)

Changing the sign of the m quantum numbers also gives a phase:

\( \begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. \)

Regge symmetries also give

\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\ j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3 \end{pmatrix}. \)
\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} \frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\ j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2} \end{pmatrix}. \)

Regge symmetries account for a total of 72 symmetries.[1] These are best displayed by the definition of a Regge symbol which is a one to one correspondence between it and a 3j symbol and assumes the properties of a semi-magic square[2]

\( R= \begin{array}{|ccc|} \hline -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\ j_1-m_1 & j_2-m_2 & j_3-m_3\\ j_1+m_1 & j_2+m_2 & j_3+m_3\\ \hline \end{array} \)

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

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

\( m_1+m_2+m_3=0\, \)

\( j_1+j_2 + j_3\text{ is an integer} \, \text{(or an even integer if} \,m_1=m_2=m_3=0)\, \)

\( |m_i| \le j_i \, \)

\( |j_1-j_2|\le j_3 \le j_1+j_2. \, \)

Scalar invariant

The contraction of the product of three rotational states with a 3j symbol,

\( \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3} |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}, \)

is invariant under rotations.
Orthogonality relations

\( (2j+1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'\\ m_1 & m_2 & m' \end{pmatrix} =\delta_{j j'}\delta_{m m'}. \)

\( \sum_{j m} (2j+1) \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j\\ m_1' & m_2' & m \end{pmatrix} =\delta_{m_{1} m_1'}\delta_{m_{2} m_2'}. \)

Relation to spherical harmonics

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

\( \begin{align} & {} \quad \int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ & = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align} \)

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:

\( \begin{align} & {} \quad \int d{\mathbf{\hat n}}\,{}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}}) \,{}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}})\, {}_{s_3} Y_{j_3m_3}({\mathbf{\hat n}}) \\[8pt] & = \sqrt{\frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix} \end{align} \)

Recursion relations

\( \begin{align} & {} \quad -\sqrt{(l_3\mp s_3)(l_3\pm s_3+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 & s_3\pm 1 \end{pmatrix} \\ & = \sqrt{(l_1\mp s_1)(l_1\pm s_1+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 \pm 1 & s_2 & s_3 \end{pmatrix} +\sqrt{(l_2\mp s_2)(l_2\pm s_2+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 \pm 1 & s_3 \end{pmatrix} \end{align} \)

Asymptotic expressions

For \( l_1\ll l_2,l_3 a non-zero 3-j symbol has

\( \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{2l_3+1}} \)

where \( \cos(\theta) = -2m_3/(2l_3+1) and d^l_{mn} \) is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

\( \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}} \)

where \( \cos(\theta) = (m_2-m_3)/(l_2+l_3+1). \)
Other properties

\( \sum_m (-1)^{j-m} \begin{pmatrix} j & j & J\\ m & -m & 0 \end{pmatrix} = \sqrt{2j+1}~ \delta_{J0} \)

\( \frac{1}{2} \int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \, dx = \begin{pmatrix} l & l_1 & l_2 \\ 0 & 0 & 0 \end{pmatrix} ^2 \)

See also

Clebsch–Gordan coefficients
Spherical harmonics
6-j symbol
9-j symbol

References

Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento 10: 544. doi:10.1007/BF02859841.
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.

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981.
D. M. Brink and G. R. Satchler, Angular Momentum, 3rd edition, Clarendon, Oxford, 1993.
A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd edition, Princeton University Press, Princeton, 1960.
Maximon, Leonard C. (2010), "3j,6j,9j Symbols", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. (1988). Quantum Theory of Angular Momentum. World Scientific Publishing Co.
Regge, T. (1958). "Symmetry Properties of Clebsch-Gordon's Coefficients". Nuovo Cimento 10 (3): 544–545. doi:10.1007/BF02859841.
E. P. Wigner, "On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups", unpublished (1940). Reprinted in: L. C. Biedenharn and H. van Dam, Quantum Theory of Angular Momentum, Academic Press, New York (1965).
Moshinsky, Marcos (1962). "Wigner coefficients for the SU3 group and some applications". Rev. Mod. Phys. 34 (4): 813. Bibcode:1962RvMP...34..813M. doi:10.1103/RevModPhys.34.813.
Baird, G. E.; Biedenharn, L. C. (1963). "On the representation of the semisimple Lie Groups. II.". J. Math. Phys. 4: 1449. Bibcode:1963JMP.....4.1449B. doi:10.1063/1.1703926.
Swart de, J. J. (1963). "The octet model and its Glebsch-Gordan coefficients". Rev. Mod. Phys. 35 (4): 916. Bibcode:1963RvMP...35..916D. doi:10.1103/RevModPhys.35.916.
Baird, G. E.; Biedenharn, L. C. (1964). "On the representations of the semisimple Lie Groups. III. The explicit conjugation Operation for SUn". J. Math. Phys. 5: 1723. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095.
Horie, Hisashi (1964). "Representations of the symmetric group and the fractional parentage coefficients". J. Phys. Soc. Jpn. 19: 1783. doi:10.1143/JPSJ.19.1783.
P. McNamee, S. J.; Chilton, Frank (1964). "Tables of Clebsch-Gordan coefficients of SU3". Rev. Mod. Phys. 36 (4): 1005. Bibcode:1964RvMP...36.1005M. doi:10.1103/RevModPhys.36.1005.
Hecht, K. T. (1965). "SU3 recoupling and fractional parentage in the 2s-1d shell". Nucl. Phys. 62 (1): 1. Bibcode:1965NucPh..62....1H. doi:10.1016/0029-5582(65)90068-4.
Itzykson, C.; Nauenberg, M. (1966). "Unitary groups: representations and decompositions". Rev. Mod. Phys. 38 (1): 95. Bibcode:1966RvMp...38...95I. doi:10.1103/RevModPhys.38.95.
Kramer, P. (1967). "Orbital fractional parentage coefficients for the harmonic oscillator shell model". Z. Physik 205 (2): 181. Bibcode:1967ZPhy..205..181K. doi:10.1007/BF01333370.
Kramer, P. (1968). "Recoupling coefficients of the symmetric group for shell and cluster model configurations". Z. Physik 216 (1): 68. Bibcode:1968ZPhy..216...68K. doi:10.1007/BF01380094.
Hecht, K. T.; Pang, Sing Ching (1969). "On the Wigner Supermultiplet Scheme". J. Math. Phys. 10 (9): 1571. Bibcode:1969JMP....10.1571H. doi:10.1063/1.1665007.
Lezuo, K. J. (1972). "The symmetric group and the Gel'fand basis of U(3). Generalizations of the Dirac identity". J. Math. Phys. 13 (9): 1389. Bibcode:1972JMP....13.1389L. doi:10.1063/1.1666151.
Draayer, J. P.; Akiyama, Yoshimi (1973). "Wigner and Racah coefficients for SU3". J. Math. Phys. 14 (12): 1904. Bibcode:1973JMP....14.1904D. doi:10.1063/1.1666267.
Akiyama, Yoshimi; Draayer, J. P. (1973). "A users' guide to fortran programs for Wigner and Racah coefficients of SU3". Comp. Phys. Comm. 5: 405. Bibcode:1973CoPhC...5..405A. doi:10.1016/0010-4655(73)90077-5.
Paldus, Josef (1974). "Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems". J. Chem. Phys 61 (12): 5321. Bibcode:1974JChPh..61.5321P. doi:10.1063/1.1681883.
Schulten, Klaus; Gordon, Roy G. (1975). "Exact recursive evaluation of 3j and 6j-coefficients for quantum mechanical coupling of angular momenta". J. Math. Phys. 16 (10): 1961–1970. Bibcode:1975JMP....16.1961S. doi:10.1063/1.522426.
Haacke, E. M.; Moffat, J. W.; Savaria, P. (1976). "A calculation of SU(4) Glebsch-Gordan coefficients". J. Math. Phys. 17 (11): 2041. Bibcode:1976JMP....17.2041H. doi:10.1063/1.522843.
Paldus, Josef (1976). "Unitary-group approach to the many-electron correlation problem: Relation of Gelfand and Weyl tableau formulations". Phys. Rev. A 14 (5): 1620. Bibcode:1976PhRvA..14.1620P. doi:10.1103/PhysRevA.14.1620.
Bickerstaff, R. P.; Butler, P. H.; Butts, M. B.; Haase, R. w.; Reid, M. F. (1982). "3jm and 6j tables for some bases of SU6 and SU3". J. Phys. A 15: 1087. Bibcode:1982JPhA...15.1087B. doi:10.1088/0305-4470/15/4/014.
Sarma, C. R.; Sahasrabudhe, G. G. (1980). "Permutational symmetry of many particle states". J. Math. Phys. 21 (4): 638. Bibcode:1980JMP....21..638S. doi:10.1063/1.524509.
Chen, Jin-Quan; Gao, Mei-Juan (1982). "A new approach to permutation group representation". J. Math. Phys. 23: 928. Bibcode:1982JMP....23..928C. doi:10.1063/1.525460.
Sarma, C. R. (1982). "Determination of basis for the irreducible representations of the unitary group for U(p+q)↓U(p)×U(q)". J. Math. Phys. 23 (7): 1235. Bibcode:1982JMP....23.1235S. doi:10.1063/1.525507.
Chen, J.-Q.; Chen, X.-G. (1983). "The Gel'fand basis and matrix elements of the graded unitary group U(m/n)". J. Phys. A 16 (15): 3435. Bibcode:1983JPhA...16.3435C. doi:10.1088/0305-4470/16/15/010.
Nikam, R. S.; Dinesha, K. V.; Sarma, C. R. (1983). "Reduction of inner-product representations of unitary groups". J. Math. Phys. 24 (2): 233. Bibcode:1983JMP....24..233N. doi:10.1063/1.525698.
Chen, Jin-Quan; Collinson, David F.; Gao, Mei-Juan (1983). "Transformation coefficients of permutation groups". J. Math. Phys. 24: 2695. Bibcode:1983JMP....24.2695C. doi:10.1063/1.525668.
Chen, Jin-Quan; Gao, Mei-Juan; Chen, Xuan-Gen (1984). "The Clebsch-Gordan coefficient for SU(m/n) Gel'fand basis". J. Phys. A 17 (3): 481. Bibcode:1984JPhA...17..727K. doi:10.1088/0305-4470/17/3/011.
Srinivasa Rao, K. (1985). "Special topics in the quantum theory of angular momentum". Pramana 24 (1): 15–26. Bibcode:1985Prama..24...15R. doi:10.1007/BF02894812.
Wei, Liqiang (1999). "Unified approach for exact calculation of angular momentum coupling and recoupling coefficients". Comp. Phys. Comm. 120 (2–3): 222–230. Bibcode:1999CoPhC.120..222W. doi:10.1016/S0010-4655(99)00232-5.
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". (Numerical)
Stevenson, Paul. "Clebsch-O-Matic". 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)

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