3-j symbol

In quantum mechanics, the Wigner 3-j symbols, also called 3j or 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically, and thus have greater and simpler symmetry properties than the Clebsch-Gordan coefficients.

Mathematical relation to Clebsch-Gordan coefficients

The 3-j symbols are given in terms of the Clebsch-Gordon 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 's and m 's 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 hand side, 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 C-G 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 C-G 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}.

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.^[1] 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^[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. These facts can be used to devise an effective storage scheme.^[3]

Orthogonality relations

A system of two angular momenta with magnitudes $j_1$ and $j_2$ , say, 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+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 _{{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 3-jm symbols give the integral of the products of three spherical harmonics

{\begin{aligned}&{}\quad \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}}\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{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

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 SU_n". 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. Bibcode:1964JPSJ...19.1783H. 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. Phys. 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. Phys. 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)
Johansson, H.T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)

This article is issued from Wikipedia - version of the 11/7/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.