Movatterモバイル変換

3-j symbol

(Redirected fromWigner 3-j symbols)

Inquantum mechanics, theWigner's 3-j symbols, also called 3-jm symbols, are an alternative toClebsch–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 .

Thej andm components are angular-momentum quantum numbers, i.e., everyj (and every correspondingm) is either a nonnegative integer orhalf-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 substitutionm₃ → −m₃:

\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}}.

Explicit expression

{\begin{aligned}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}&\equiv \delta (m_{1}+m_{2}+m_{3},0)(-1)^{j_{1}-j_{2}-m_{3}}{}{\sqrt {\frac {(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}}}\ \times {}\\[6pt]&\times {\sqrt {(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!(j_{3}-m_{3})!(j_{3}+m_{3})!}}\ \times {}\\[6pt]&\times \sum _{k=K}^{N}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j_{3}-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j_{3}-j_{2}+m_{1}+k)!(j_{3}-j_{1}-m_{2}+k)!}},\end{aligned}}

where $\delta (i,j)$ is theKronecker delta.

The summation is performed over those integer valuesk for which the argument of eachfactorial in the denominator is non-negative, i.e. summation limitsK andN are taken equal: the lower one $K=\max(0,j_{2}-j_{3}-m_{1},j_{1}-j_{3}+m_{2}),$ the upper one $N=\min(j_{1}+j_{2}-j_{3},j_{1}-m_{1},j_{2}+m_{2}).$ Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, $j_{3}>j_{1}+j_{2}$ or $j_{1}<m_{1}$ are automatically set to zero.

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 $m {\displaystyle m}$ 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.^[3]

Orthogonality relations

A system of two angular momenta with magnitudesj₁ andj₂ can be described either in terms of the uncoupled basis states (labeled by the quantum numbersm₁ andm₂), or the coupled basis states (labeled byj₃ andm₃). 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}'}.

Thetriangular delta{j₁ j₂ j₃} 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^[4] a "3-j symbol" (without them) in analogy to6-j and9-j symbols, all of which are irreducible summations of 3-jm symbols where nom variables remain.

Relation to spherical harmonics; Gaunt coefficients

The 3-jm symbols give the integral of the products of threespherical harmonics^[5]

{\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. These integrals are called Gaunt coefficients.

Relation to integrals of spin-weighted spherical harmonics

Similar relations exist for thespin-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 aWigner 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 ametric tensor in angular-momentum theory and is also known as aWigner 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.

Special cases and other properties

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

From equation (3.7.9) in^[6]

{\begin{pmatrix}j&j&0\\m&-m&0\end{pmatrix}}={\frac {1}{\sqrt {2j+1}}}(-1)^{j-m}.

{\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},

whereP areLegendre polynomials.

Relation to RacahV-coefficients

Wigner 3-j symbols are related toRacahV-coefficients^[7] 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}}.

Relation to group theory

This section essentially recasts thedefinitional relationin the language of group theory.

Agroup representation of agroup is ahomomorphism of the group intoa group oflinear transformations over some vector space. The lineartransformations can be given by a group of matrices with respect to some basis of the vector space.

The group of transformations leaving angular momenta invariant is the three dimensional rotation groupSO(3). When "spin" angular momenta are included, the group is itsdouble covering group,SU(2).

A reducible representation is one where a change of basis can be applied to bring all the matrices into block diagonal form. A representationisirreducible (irrep) if no such transformation exists.

For each value ofj, the 2j+1 kets form a basis for an irreducible representation (irrep) ofSO(3)/SU(2) over the complex numbers. Given twoirreps, thetensor direct product can be reduced to asum of irreps, giving rise to the Clebcsh-Gordon coefficients, or by reduction of the triple product of three irreps to thetrivial irrep1 giving rise to the 3j symbols.

3j symbols for other groups

The $3j$ symbol has been most intensely studied in the context of the coupling of angular momentum. For this, it is strongly related to thegroup representation theory of the groups SU(2) and SO(3)as discussed above. However, manyother groups are of importance inphysics andchemistry,and there has been much work on the $3j$ symbol for these other groups.In this section, some of that work is considered.

Simply reducible groups

The original paper by Wigner^[1]was not restricted to SO(3)/SU(2)but instead focussed on simply reducible (SR) groups.These are groups in which

all classes are ambivalent i.e. if $X {\displaystyle X}$ is a member of a class then so is $X^{-1}$
the Kronecker product of two irreps is multiplicity free i.e. does not contain any irrep more than once.

For SR groups, every irrep is equivalent to its complex conjugate,and under permutations of the columns the absolute value of thesymbol is invariant and the phase of each can be chosen so thatthey at most change sign under odd permutations and remainunchanged under even permutations.

General compact groups

Compact groups form a wide class of groups withtopological structure.They include the finite groups with addeddiscrete topologyand many of theLie groups.

General compact groups will neither be ambivalent nor multiplicity free.Derome and Sharp^[8]and Derome^[9] examined the $3j$ symbolfor the general case using the relation to the Clebsch-Gordon coefficients of

{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {1}{[j_{3}]}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}^{*}\,m_{3}\rangle .

where $[j]$ is the dimension of the representation space of $j {\displaystyle j}$ and $j_{3}^{*}$ is the complex conjugaterepresentation to $j_{3}$ .

By examining permutations of columns of the $3j$ symbol, they showed three cases:

if all of $j_{1},j_{2},j_{3}$ are inequivalent then the $3j$ symbol may be chosen to be invariant under any permutation of its columns
if exactly two are equivalent, then transpositions of its columns may be chosen so that some symbols will be invariant while others will change sign. An approach using awreath product of the group with $S_{3}$ ^[10] showed that these correspond to therepresentations $[2] {\displaystyle [2]}$ or $[1^{2}]$ of the symmetric group $S_{2}$ . Cyclic permutations leave the $3j$ symbol invariant.
if all three are equivalent, the behaviour is dependent on therepresentations of the symmetric group $S_{3}$ . Wreath group representations corresponding to $[3] {\displaystyle [3]}$ are invariant under transpositions of the columns, corresponding to $[1^{3}]$ change sign under transpositions, while a pair corresponding to the two dimensional representation $[21] {\displaystyle [21]}$ transform according to that.

Further research into $3j$ symbols for compact groups has been performed based on these principles.^[11]

SU(n)

TheSpecial unitary group SU(n) is theLie group of n × n unitary matrices with determinant 1.

The groupSU(3) is important inparticle theory.There are many papers dealing with the $3j$ orequivalent symbol^[12]^[13]^[14]^[15]^[16]^[17]^[18]^[19]

The $3j$ symbol for the group SU(4) has been studied^[20]^[21]while there is also work on the general SU(n) groups^[22]^[23]

Crystallographic point groups

There are many papers dealing with the $3j$ symbols or Clebsch-Gordon coefficients for the finitecrystallographic point groupsand thedouble point groupsThe book by Butler^[24]references these and details the theory along with tables.

Magnetic groups

Magnetic groups include antilinear operators as well as linear operators. They need to be dealt with usingWigner's theory ofcorepresentations of unitary and antiunitary groups.A significant departure from standard representation theory is that the multiplicity of the irreducible corepresentation $j_{3}^{*}$ in the direct product of the irreducible corepresentations $j_{1}\otimes j_{2}$ is generally smaller than the multiplicity of the trivial corepresentation in the tripleproduct $j_{1}\otimes j_{2}\otimes j_{3}$ , leading to significant differences between the Clebsch-Gordoncoefficients and the $3j$ symbol.

The $3j$ symbols have been examined for the grey groups^[25]^[26]and for the magnetic point groups^[27]

See also

References

^^a ^b ^cWigner, E. P. (1993). "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. Vol. A/1. pp. 608–654.doi:10.1007/978-3-662-02781-3_42.ISBN 978-3-642-08154-5.
^Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients".Nuovo Cimento.10 (3): 544.Bibcode:1958NCim...10..544R.doi:10.1007/BF02859841.S2CID 122299161.
^^a ^bRasch, 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.51. Elsevier:59–124.Bibcode:2006AdQC...51...59W.doi:10.1016/S0065-3276(06)51002-0.ISBN 9780120348510.ISSN 0065-3276.
^Cruzan, Orval R. (1962)."Translational addition theorems for spherical vector wave functions".Quarterly of Applied Mathematics.20 (1):33–40.doi:10.1090/qam/132851.ISSN 0033-569X.
^Edmonds, Alan (1957).Angular Momentum in Quantum Mechanics. Princeton University Press.
^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.
^Derome, J-R; Sharp, W. T. (1965). "Racah Algebra for an Arbitrary Group".J. Math. Phys.6 (10):1584–1590.Bibcode:1965JMP.....6.1584D.doi:10.1063/1.1704698.
^Derome, J-R (1966). "Symmetry Properties of the 3j Symbols for an Arbitrary Group".J. Math. Phys.7 (4):612–615.Bibcode:1966JMP.....7..612D.doi:10.1063/1.1704973.
^Newmarch, J. D. (1983). "On the 3j symmetries".J. Math. Phys.24 (4):757–764.Bibcode:1983JMP....24..757N.doi:10.1063/1.525771.
^Butler, P. H.; Wybourne, B. G. (1976). "Calculation ofj andjm Symbols forArbitrary Compact Groups. I. Methodology".Int. J. Quantum Chem.X (4):581–598.doi:10.1002/qua.560100404.
^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.
^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.
^Draayer, J. P.; Akiyama, Yoshimi (1973)."Wigner and Racah coefficients for SU₃"(PDF).J. Math. Phys.14 (12): 1904.Bibcode:1973JMP....14.1904D.doi:10.1063/1.1666267.hdl:2027.42/70151.
^Akiyama, Yoshimi; Draayer, J. P. (1973). "A users' guide to fortran programs for Wigner and Racah coefficients of SU₃".Comput. Phys. Commun.5 (6): 405.Bibcode:1973CoPhC...5..405A.doi:10.1016/0010-4655(73)90077-5.hdl:2027.42/24983.
^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 (4): 1087.Bibcode:1982JPhA...15.1087B.doi:10.1088/0305-4470/15/4/014.
^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.
^Derome, J-R (1967). "Symmetry Properties of the 3j Symbols for SU(3)".J. Math. Phys.8 (4):714–716.Bibcode:1967JMP.....8..714D.doi:10.1063/1.1705269.
^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.hdl:2027.42/32049.
^Hecht, K. T.; Pang, Sing Ching (1969)."On the Wigner Supermultiplet Scheme"(PDF).J. Math. Phys.10 (9): 1571.Bibcode:1969JMP....10.1571H.doi:10.1063/1.1665007.hdl:2027.42/70485.
^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.
^Baird, G. E.; Biedenharn, L. C. (1963). "On the representation of the semisimple Lie Groups. II".J. Math. Phys.4 (12): 1449.Bibcode:1963JMP.....4.1449B.doi:10.1063/1.1703926.
^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 (12): 1723.Bibcode:1964JMP.....5.1723B.doi:10.1063/1.1704095.
^Butler, P. H. (1981).Point Group Symmetry Applications: methods and tables. Plenum Press, New York.
^Newmarch, J. D. (1981).The Racah Algebra for Groups with Time Reversal Symmetry (Thesis). University of New South Wales.
^Newmarch, J. D.; Golding, R. M. (1981)."The Racah Algebra for Groups with Time Reversal Symmetry".J. Math. Phys.22 (2):233–244.Bibcode:1981JMP....22..233N.doi:10.1063/1.524894.hdl:1959.4/69692.
^Kotsev, J. N.; Aroyo, M. I.; Angelova, M. N. (1984). "Tables of Spectroscopic Coefficients for Magnetic Point Group Symmetry".J. Mol. Structure.115:123–128.doi:10.1016/0022-2860(84)80030-7.

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", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,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.Bibcode:1958NCim...10..544R.doi:10.1007/BF02859841.S2CID 122299161.

Horie, Hisashi (1964). "Representations of the symmetric group and the fractional parentage coefficients".J. Phys. Soc. Jpn.19 (10): 1783.Bibcode:1964JPSJ...19.1783H.doi:10.1143/JPSJ.19.1783.
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.OSTI 1444219.
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.S2CID 122879812.
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.S2CID 121508850.
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.
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.
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.
Raynal, Jacques (1978). "On the definition and properties of generalized 3-j symbols".J. Math. Phys.19 (2): 467.doi:10.1063/1.523668.
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 (6): 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 (12): 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.S2CID 120663002.
Wei, Liqiang (1999). "Unified approach for exact calculation of angular momentum coupling and recoupling coefficients".Comput. Phys. Commun.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". Archived fromthe 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)

Retrieved from "https://en.wikipedia.org/w/index.php?title=3-j_symbol&oldid=1257743198"

[8]ページ先頭

©2009-2025 Movatter.jp