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词条 3-j symbol
释义

  1. Mathematical relation to Clebsch-Gordan coefficients

  2. Definitional relation to Clebsch-Gordan coefficients

  3. Selection rules

  4. Symmetry properties

  5. Orthogonality relations

  6. Relation to spherical harmonics

      Relation to integrals of spin-weighted spherical harmonics  

  7. Recursion relations

  8. Asymptotic expressions

  9. Metric tensor

  10. Other properties

  11. Relation to Racah {{math|V}}-coefficients

  12. See also

  13. References

  14. External links

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, 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-Gordan coefficients by

The j 's and m 's are angular momentum quantum numbers, i.e., every {{math|j}} (and every corresponding {{math|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 {{math|m3 → −m3}}:

.

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:

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:

Here, is the zero angular momentum state (). 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 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:

Symmetry properties

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

An odd permutation of the columns gives a phase factor:

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

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

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]

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 {{math|j1}} and {{math|j2}}, say, can be described either in terms of the uncoupled basis states (labeled by the quantum numbers {{math|m1}} and {{math|m2}}), or the coupled basis states (labeled by {{math|j3}} and {{math|m3}}). The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations,

The triangular delta {{math|{j1  j2  j3}}} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and 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 {{math|m}} variables remain.

Relation to spherical harmonics

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

with , and integers.

Relation to integrals of spin-weighted spherical harmonics

Similar relations exist for the spin-weighted spherical harmonics if :

Recursion relations

Asymptotic expressions

For a non-zero 3-j symbol has

where and is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

where .

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]

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

Other properties

Relation to Racah {{math|V}}-coefficients

Wigner 3-j symbols are related to Racah {{math|V}}-coefficients[6] by a simple phase:

See also

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

References

1. ^1 {{cite book |last=Wigner |first=E. P. |editor1-last=Wightman |editor1-first=Arthur S. |date=1951 |chapter=On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups |title=The Collected Works of Eugene Paul Wigner |volume=3 |pages=608–654 |doi=10.1007/978-3-662-02781-3_42 |isbn=978-3-642-08154-5 }}
2. ^{{cite journal |first1=T. |last1=Regge|title=Symmetry Properties of Clebsch-Gordan Coefficients |journal=Nuovo Cimento |year=1958|volume=10 |issue=3|page=544 |doi=10.1007/BF02859841|bibcode=1958NCim...10..544R}}
3. ^{{Cite journal |first1=J. |last1=Rasch|first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932}}
4. ^{{Cite journal |first1=J. |last1=Rasch |first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932}}
5. ^{{cite article|title=Angular Momentum Diagrams|author1=P.E.S. Wormer |author2=J. Paldus |journal=Advances in Quantum Chemistry|publisher=Elsevier|volume=51|pages=59–124|year=2006|issn=0065-3276|doi=10.1016/S0065-3276(06)51002-0|url=http://www.sciencedirect.com/science/article/pii/S0065327606510020|bibcode = 2006AdQC...51...59W }}
6. ^{{Cite journal |first=G. |last=Racah |title=Theory of Complex Spectra II |journal=Physical Review |volume=62 |issue=9–10 |pages=438–462 |year=1942 |doi=10.1103/PhysRev.62.438 |bibcode = 1942PhRv...62..438R }}
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.
  • {{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
  • {{cite book

|first1=D. A.
|last1=Varshalovich
|first2=A. N.
|last2=Moskalev
|first3=V. K.
|last3=Khersonskii
|title=Quantum Theory of Angular Momentum
|publisher=World Scientific Publishing Co.
|year=1988
}}
  • {{ cite journal

|first1=T.
|last1=Regge
|title=Symmetry Properties of Clebsch-Gordon's Coefficients
|journal=Nuovo Cimento
|year=1958
|volume=10 |issue=3
|doi=10.1007/BF02859841
|pages=544–545
|bibcode=1958NCim...10..544R
}}
  • {{Cite journal

|first1=Marcos
|last1=Moshinsky
|title=Wigner coefficients for the SU3 group and some applications
|journal=Rev. Mod. Phys.
|volume=34
|year=1962
|page=813
|doi=10.1103/RevModPhys.34.813
|issue=4
|bibcode = 1962RvMP...34..813M }}
  • {{Cite journal

|first1=G. E.
|last1=Baird
|first2=L. C.
|last2=Biedenharn
|title=On the representation of the semisimple Lie Groups. II
|journal=J. Math. Phys.
|volume=4
|issue=12
|year=1963
|page=1449
|doi=10.1063/1.1703926
|bibcode=1963JMP.....4.1449B
}}
  • {{Cite journal

|first1=J. J.
|last1=Swart de
|title=The octet model and its Glebsch-Gordan coefficients
|journal=Rev. Mod. Phys.
|volume=35
|year=1963
|page=916
|doi=10.1103/RevModPhys.35.916
|issue=4
|bibcode = 1963RvMP...35..916D |url=http://cds.cern.ch/record/345262
}}
  • {{Cite journal

|first1=G. E.
|last1=Baird
|first2=L. C.
|last2=Biedenharn
|title=On the representations of the semisimple Lie Groups. III. The explicit conjugation Operation for SUn
|journal=J. Math. Phys.
|volume=5
|issue=12
|year=1964
|page=1723
|doi=10.1063/1.1704095
|bibcode = 1964JMP.....5.1723B }}
  • {{Cite journal

|first1=Hisashi
|last1=Horie
|title=Representations of the symmetric group and the fractional parentage coefficients
|journal=J. Phys. Soc. Jpn.
|volume=19
|issue=10
|year=1964
|page=1783
|doi=10.1143/JPSJ.19.1783
|bibcode = 1964JPSJ...19.1783H }}
  • {{Cite journal

|first1=S. J.
|last1=P. McNamee
|first2=Frank
|last2=Chilton
|title=Tables of Clebsch-Gordan coefficients of SU3
|journal=Rev. Mod. Phys.
|volume=36
|year=1964
|page=1005
|doi=10.1103/RevModPhys.36.1005
|issue=4
|bibcode=1964RvMP...36.1005M
}}
  • {{Cite journal

|first1=K. T.
|last1=Hecht
|title=SU3 recoupling and fractional parentage in the 2s-1d shell
|journal=Nucl. Phys.
|volume=62
|year=1965
|page=1
|doi=10.1016/0029-5582(65)90068-4
|issue=1
|bibcode=1965NucPh..62....1H
}}
  • {{Cite journal

|first1=C.
|last1=Itzykson
|first2=M.
|last2=Nauenberg
|title=Unitary groups: representations and decompositions
|journal=Rev. Mod. Phys.
|volume=38
|year=1966
|page=95
|doi=10.1103/RevModPhys.38.95
|issue=1
|bibcode=1966RvMp...38...95I
}}
  • {{Cite journal

|first1=P.
|last1=Kramer
|title=Orbital fractional parentage coefficients for the harmonic oscillator shell model
|journal=Z. Phys.
|volume=205
|year=1967
|page=181
|doi=10.1007/BF01333370
|issue=2
|bibcode = 1967ZPhy..205..181K }}
  • {{Cite journal

|first1=P.
|last1=Kramer
|title=Recoupling coefficients of the symmetric group for shell and cluster model configurations
|journal=Z. Phys.
|volume=216
|year=1968
|page=68
|doi=10.1007/BF01380094
|issue=1
|bibcode = 1968ZPhy..216...68K }}
  • {{Cite journal

|first1=K. T.
|last1=Hecht
|first2=Sing Ching
|last2=Pang
|title=On the Wigner Supermultiplet Scheme
|journal=J. Math. Phys.
|volume=10
|year=1969
|page=1571
|doi=10.1063/1.1665007
|issue=9
|bibcode = 1969JMP....10.1571H }}
  • {{Cite journal

|first1=K. J.
|last1=Lezuo
|title=The symmetric group and the Gel'fand basis of U(3). Generalizations of the Dirac identity
|journal=J. Math. Phys.
|volume=13
|year=1972
|page=1389
|doi=10.1063/1.1666151
|issue=9
|bibcode = 1972JMP....13.1389L }}
  • {{Cite journal

|first1=J. P.
|last1=Draayer
|first2=Yoshimi
|last2=Akiyama
|title=Wigner and Racah coefficients for SU3
|journal=J. Math. Phys.
|volume=14
|year=1973
|page=1904
|doi=10.1063/1.1666267
|issue=12
|bibcode = 1973JMP....14.1904D |url=https://deepblue.lib.umich.edu/bitstream/2027.42/70151/2/JMAPAQ-14-12-1904-1.pdf
}}
  • {{Cite journal

|first1=Yoshimi
|last1=Akiyama
|first2=J. P.
|last2=Draayer
|title=A users' guide to fortran programs for Wigner and Racah coefficients of SU3
|journal=Comp. Phys. Comm.
|volume=5
|issue=6
|year=1973
|page=405
|doi=10.1016/0010-4655(73)90077-5
|bibcode = 1973CoPhC...5..405A }}
  • {{Cite journal

|first1=Josef
|last1=Paldus
|title=Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems
|journal=J. Chem. Phys.
|volume=61
|year=1974
|page=5321
|doi=10.1063/1.1681883
|issue=12
|bibcode = 1974JChPh..61.5321P }}
  • {{cite journal

|first1=Klaus
|last1=Schulten
|first2=Roy G.
|last2=Gordon
|title=Exact recursive evaluation of 3j and 6j-coefficients for quantum mechanical coupling of angular momenta
|journal=J. Math. Phys.
|volume=16
|issue=10
|pages=1961–1970
|year=1975
|doi=10.1063/1.522426
|bibcode = 1975JMP....16.1961S }}
  • {{Cite journal

|first1=E. M.
|last1=Haacke
|first2=J. W.
|last2=Moffat
|first3=P.
|last3=Savaria
|title=A calculation of SU(4) Glebsch-Gordan coefficients
|journal=J. Math. Phys.
|volume=17
|year=1976
|page=2041
|doi=10.1063/1.522843
|issue=11
|bibcode = 1976JMP....17.2041H }}
  • {{Cite journal

|first1=Josef
|last1=Paldus
|title=Unitary-group approach to the many-electron correlation problem: Relation of Gelfand and Weyl tableau formulations
|journal=Phys. Rev. A
|volume=14
|year=1976
|page=1620
|doi=10.1103/PhysRevA.14.1620
|issue=5
|bibcode = 1976PhRvA..14.1620P }}
  • {{Cite journal

|first1=R. P.
|last1=Bickerstaff
|first2=P. H.
|last2=Butler
|first3=M. B.
|last3=Butts
|first4=R. w.
|last4=Haase
|first5=M. F.
|last5=Reid
|title=3jm and 6j tables for some bases of SU6 and SU3
|journal=J. Phys. A
|volume=15
|issue=4
|year=1982
|page=1087
|doi=10.1088/0305-4470/15/4/014
|bibcode=1982JPhA...15.1087B
}}
  • {{Cite journal

|first1=C. R.
|last1=Sarma
|first2=G. G.
|last2=Sahasrabudhe
|title=Permutational symmetry of many particle states
|journal=J. Math. Phys.
|volume=21
|year=1980
|page=638
|doi=10.1063/1.524509
|issue=4
|bibcode=1980JMP....21..638S
}}
  • {{Cite journal

|first1=Jin-Quan
|last1=Chen
|first2=Mei-Juan
|last2=Gao
|title=A new approach to permutation group representation
|journal=J. Math. Phys.
|volume=23
|issue=6
|year=1982
|page=928
|doi=10.1063/1.525460
|bibcode=1982JMP....23..928C
}}
  • {{Cite journal

|first1=C. R.
|last1=Sarma
|title=Determination of basis for the irreducible representations of the unitary group for U(p+q)↓U(p)×U(q)
|journal=J. Math. Phys.
|volume=23
|year=1982
|page=1235
|doi=10.1063/1.525507
|issue=7
|bibcode=1982JMP....23.1235S
}}
  • {{Cite journal

|first1=J.-Q.
|last1=Chen
|first2=X.-G.
|last2=Chen
|title=The Gel'fand basis and matrix elements of the graded unitary group U(m/n)
|journal=J. Phys. A
|volume=16
|year=1983
|page=3435
|doi=10.1088/0305-4470/16/15/010
|issue=15
|bibcode = 1983JPhA...16.3435C }}
  • {{Cite journal

|first1=R. S.
|last1=Nikam
|first2=K. V.
|last2=Dinesha
|first3=C. R.
|last3=Sarma
|title=Reduction of inner-product representations of unitary groups
|journal=J. Math. Phys.
|volume=24
|year=1983
|page=233
|doi=10.1063/1.525698
|issue=2
|bibcode = 1983JMP....24..233N }}
  • {{Cite journal

|first1=Jin-Quan
|last1=Chen
|first2=David F.
|last2=Collinson
|first3=Mei-Juan
|last3=Gao
|title=Transformation coefficients of permutation groups
|journal=J. Math. Phys.
|volume=24
|issue=12
|year=1983
|page=2695
|doi=10.1063/1.525668
|bibcode=1983JMP....24.2695C
}}
  • {{Cite journal

|first1=Jin-Quan
|last1=Chen
|first2=Mei-Juan
|last2=Gao
|first3=Xuan-Gen
|last3=Chen
|title=The Clebsch-Gordan coefficient for SU(m/n) Gel'fand basis
|journal=J. Phys. A
|volume=17
|year=1984
|page=481
|doi=10.1088/0305-4470/17/3/011
|issue=3
|bibcode=1984JPhA...17..727K
}}
  • {{ cite journal

|first1=K.
|last1=Srinivasa Rao
|title=Special topics in the quantum theory of angular momentum
|year=1985
|volume=24 | number=1 | pages=15–26
|journal=Pramana
|doi=10.1007/BF02894812
|bibcode = 1985Prama..24...15R }}
  • {{ cite journal

|first1=Liqiang
|last1=Wei
|title=Unified approach for exact calculation of angular momentum coupling and recoupling coefficients
|journal=Comp. Phys. Comm.
|year=1999
|volume=120
|issue=2–3
|pages=222–230
|doi=10.1016/S0010-4655(99)00232-5
|bibcode=1999CoPhC.120..222W
}}
  • {{Cite journal

|first1=J.
|last1=Rasch
|first2=A. C. H.
|last2=Yu
|title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients
|journal=SIAM J. Sci. Comput.
|volume=25
|issue=4
|year=2003
|pages=1416–1428
|doi=10.1137/s1064827503422932
}}

External links

  • {{cite web

|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator
}}
  • {{cite web

|first1 = A.
|last1 = Volya
|url = http://www.volya.net/vc/vc.php
|archive-url = https://web.archive.org/web/20070929011249/http://www.volya.net/vc/vc.php
|dead-url = yes
|archive-date = 2007-09-29
|title = Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator

}} (Numerical)

  • {{cite journal

|first1=Paul
|last1=Stevenson
|url=http://personal.ph.surrey.ac.uk/~phs3ps/cleb.html
|title=Clebsch-O-Matic
|journal=Computer Physics Communications
|volume=147
|issue=3
|pages=853–858
|doi=10.1016/S0010-4655(02)00462-9
|bibcode = 2002CoPhC.147..853S |year=2002
}}
  • 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
  • {{cite web

|first1=H.T.
|last1=Johansson
|first2=C.
|last2=Forssén
|title=(WIGXJPF)
|url=http://fy.chalmers.se/subatom/wigxjpf/

}} (accurate; C, fortran, python)

  • {{cite web

|first1=H.T.
|last1=Johansson
|title=(FASTWIGXJ)
|url=http://fy.chalmers.se/subatom/fastwigxj/

}} (fast lookup, accurate; C, fortran)

3 : Rotational symmetry|Representation theory of Lie groups|Quantum mechanics
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