3j-символи

In quantum mechanics, the Wigner 3-jm symbols, also called 3j 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 number 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:

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

Selection rules

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

$m_{1}+m_{2}+m_{3}=0\,$

$j_{1}+j_{2}+j_{3}\,$ is integer

$|m_{i}|\leq j_{i}$

$|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2}$.

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

$(2j+1)\sum _{m_{1}m_{2}}{\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}&m_{2}&m\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j'\\m_{1}&m_{2}&m'\end{pmatrix}}=\delta _{jj'}\delta _{mm'}.$

$\sum _{jm}(2j+1){\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}&m_{2}&m\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}'&m_{2}'&m\end{pmatrix}}=\delta _{m_{1}m_{1}'}\delta _{m_{2}m_{2}'}.$

Relation to spherical harmonics

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

$\int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi ={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}$

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

Relation to integrals of spin-weighted spherical harmonics

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

Other properties

$\sum _{m}(-1)^{j+m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {\frac {2j+1}{2J+1}}}\delta _{J0}$

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

Див.також

• Clebsch-Gordan coefficients
• Spherical harmonics
• 6j символи
• 9j символи
• 12j символи
• 15j символи

Джерела

• 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.
• Шаблон:Dlmf
• D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific Publishing Co., Singapore, 1988.
• 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. 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. doi:10.1063/1.1703926.
• Swart, J. J. (1963). The octet model and its Glebsch-Gordan coefficients. Rev. Mod. Phys. 35 (4): 916. 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 SU_n. J. Math. Phys. 5: 1723. 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; Chilton, Frank (1964). Tables of Clebsch-Gordan coefficients of SU₃. Rev. Mod. Phys. 36 (4): 1005. 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. doi:10.1016/0029-5582(65)90068-4.
• Itzykson, C.; Nauenberg, M. (1966). Unitary groups: representations and decompositions. Rev. Mod. Phys. 38 (1): 95. doi:10.1103/RevModPhys.38.95.
• Kramer, P. (1967). Orbital fractional parentage coefficients for the harmonic oscillator shell model. Z. Physik. 205 (2): 181. doi:10.1007/BF01333370.
• Kramer, P. (1968). Recoupling coefficients of the symmetric group for shell and cluster model configurations. Z. Physik. 216 (1): 68. doi:10.1007/BF01380094.
• Hecht, K. T.; Pang, Sing Ching (1969). On the Wigner Supermultiplet Scheme. J. Math. Phys. 10 (9): 1571. 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. doi:10.1063/1.1666151.
• Draayer, J. P.; Akiyama, Yoshimi (1973). Wigner and Racah coefficients for SU₃. J. Math. Phys. 14 (12): 1904. 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. 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. doi:10.1063/1.1681883.
• Haacke, E. M.; Moffat, J. W.; Savaria, P. (1976). A calculation of SU(4) Glebsch-Gordan coefficients. J. Math. Phys. 17 (11): 2041. 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. 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. 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. doi:10.1063/1.524509.
• Chen, Jin-Quan; Gao, Mei-Juan (1982). A new approach to permutation group representation. J. Math. Phys. 23: 928. doi:10.1063/1.525460.
• Sarma, C. R.; Sarma (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. 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. 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. 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. 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. doi:10.1088/0305-4470/17/3/011.

Зовнішні посилання

• Anthony Stone’s Wigner coefficient calculator (Gives exact answer)
• Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator (Numerical)
• 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)