3j symbol
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Abstract
In quantum mechanics, the Wigner 3j symbols, also called 3j or 3jm 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 3j symbols do so more symmetrically, and thus have greater and simpler symmetry properties than the ClebschGordan coefficients.
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Related papers
Semiclassical mechanics of the Wigner 6 j -symbol
Journal of Physics A: Mathematical and Theoretical, 2012
The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol. PACS numbers: 03.65.Sq, 02.20.Qs, 02.30.Ik, 02.40.Yy I r J r J 2 12 J 2 12 J 2 23 J 2 23 J tot 0 operator Weyl symbol I r I r − 1/2 J r J r J tot J tot
Semiclassical analysis of the Wigner 9j symbol with small and large angular momenta
Physical Review A, 2011
We derive a new asymptotic formula for the Wigner 9j-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant non-canonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher 3nj-symbols. We display without proof some new asymptotic formulas for the 12j-symbol and the 15j-symbol in the appendices. This work contributes a new asymptotic formula of the Wigner 9j-symbol to the quantum theory of angular momentum, and serves as an example of a new general method for deriving asymptotic formulas for 3nj-symbols.
Exact computation and asymptotic approximations of 6 j symbols: Illustration of their semiclassical limits
International Journal of Quantum Chemistry, 2010
This article describes a direct method for the exact computation of 3nj symbols from the defining series, and continues discussing properties and asymptotic formulas focusing on the most important case, the 6j symbols or Racah coefficients. Relationships with families of hypergeometric orthogonal polynomials are presented and the asymptotic behavior is studied to account for some of the most relevant features, both from the viewpoints of the basic geometrical significance and as a source of accurate approximation formulas, such as those due to Ponzano and Regge and Schulten and Gordon. Numerical aspects are specifically investigated in detail, regarding the relationship between functions of discrete and of continuous variables, exhibiting the transition in the limit of large angular momenta toward both Wigner's reduced rotation matrices (or Jacobi polynomials) and harmonic oscillators (or Hermite polynomials).
Zeros of 6-j symbols: Atoms, nuclei, and bosons
Physical Review C, 2011
The Edmonds asymptotic formulas for the 3j and 6j symbols
Journal of Mathematical Physics, 1998
The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulae given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.
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The Edmonds asymptotic formulae for the 3j and 6j symbols
arXiv (Cornell University), 1997
The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulae given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.
Angular Momentum Coupling and Discrete Quantum Mechanics: the 10-Spin Network, the Pentagonal Relationship, an Eigenvalue Equation and Semiclassical Limits
Revista processos químicos, 2015
Racah - Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams, and quantum groupoids
2015
We relate quantum 6J symbols of various types (quantum versions of Wigner and Racah symbols) to Ocneanu cells associated with AN Dynkin diagrams. We check explicitly the algebraic structure of the associated quantum groupoids and analyze several examples. Some features relative to cells associated with more general ADE diagrams are also discussed.
Wigner and Racah coefficients for SU3
Journal of Mathematical Physics, 1973
A general yet simple and hence practical algorithm for calculating SV, ::l SV 2 X V, Wigner coefficients is formulated. The resolution of the outer multiplicity follows the prescription given by Biedenharn and Louck. Ii is shown that SV 3 Racah coefficients can be obtained as a solution to a set of simultaneous equations with unknown coefficients given as a by-product of the initial steps in the SV3 ::l SV2 X VI Wigner coefficient construction algorithm. A general expression for evaluating SV3 ::l R3 Wigner coefficients as a sum over a simple subset of the corresponding SU 3 ::l SV 2 X VI Wigner coefficients is also presented. State conjugation properties are discussed and symmetry relations for both the SV3::l SV 2 X VI and SV 3 ::l R3 Wigner coefficients are given. Machine codes based on the results are available.
Multiplicity-free Quantum 6 j-Symbols for Uq(slN )
2013
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6 j-symbols for Uq (slN ). The expression is a natural generalization of the quan- tum 6 j-symbols for Uq (sl2) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carry- ing arbitrary symmetric representations.
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3j symbol Wikipedia
3j symbol
From →ikipedia, the free encyclopedia
In quantum mechanics, the Wigner 3j symbols, also called 3j or 3jm symbols, are an alternative to
Clebsch–Gordan coefficients for the purpose of adding angular momenta. →hile the two approaches address
exactly the same physical problem, the 3j symbols do so more symmetrically, and thus have greater and
simpler symmetry properties than the ClebschGordan coefficients.
Contents
1
2
3
4
5
6
Mathematical relation to ClebschGordan coefficients
Definitional relation to ClebschGordan coefficients
Selection rules
Symmetry properties
Orthogonality relations
Relation to spherical harmonics
6.1 Relation to integrals of spinweighted spherical harmonics
7 Recursion relations
8 Asymptotic expressions
9 Other properties
10 See also
11 References
12 External links
Mathematical relation to ClebschGordan coefficients
The 3j symbols are given in terms of the ClebschGordon coefficients by
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 halfoddinteger. 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
m3
−m3:
.
Definitional relation to ClebschGordan coefficients
The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third:
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The 3j 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 3j symbol treats all three
angular momenta involved in the addition problem on an equal footing, and is therefore more symmetrical
than the CG coefficient.
is unchanged by rotation, one also says that the contraction of the product of three
Since the state
rotational states with a 3j symbol is invariant under rotations.
Selection rules
The →igner 3j symbol is zero unless all these conditions are satisfied:
Symmetry properties
A 3j 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 (timereversal) also gives a phase:
The 3j symbols also have socalled Regge symmetries, which are not due to permutations or time
reversal.[1] These symmetries are,
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→ith the Regge symmetries, the 3j symbol has a total of 72 symmetries. These are best displayed by the
definition of a Regge symbol which is a onetoone correspondence between it and a 3j symbol and assumes
the properties of a semimagic square[2]
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 and , say, can be described either in terms of the
and
), or the coupled basis states (labeled by
uncoupled basis states (labeled by the quantum numbers
and
). The 3j symbols constitute a unitary transformation between these two bases, and this unitarity
implies the orthogonality relations,
Relation to spherical harmonics
The 3jm symbols give the integral of the products of three spherical harmonics
with
,
and
integers.
Relation to integrals of spinweighted spherical harmonics
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Similar relations exist for the spinweighted spherical harmonics if
:
Recursion relations
Asymptotic expressions
a nonzero 3j symbol has
For
where
the Regge symmetry is given by
and
where
is a →igner function. Generally a better approximation obeying
.
Other properties
See also
Clebsch–Gordan coefficients
Spherical harmonics
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3j symbol Wikipedia
6j symbol
9j symbol
References
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doi:10.1007/BF02859841.
2. Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Precalculated →igner 3j, 6j and Gaunt
Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
3. Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Precalculated →igner 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→esley, 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 →. J.; Lozier, Daniel M.; Boisvert,
Ronald F.; Clark, Charles →., NIST Handbook of Mathematical Functions, Cambridge University
Press, ISBN 9780521192255, MR 2723248
Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. (1988). Quantum Theory of Angular
Momentum. →orld Scientific Publishing Co.
Regge, T. (1958). "Symmetry Properties of ClebschGordon's Coefficients". Nuovo Cimento. 10 (3):
544–545. doi:10.1007/BF02859841.
E. P. →igner, "On the Matrices →hich Reduce the Kronecker Products of Representations of Simply
Reducible Groups", unpublished (1940). Reprinted in: L. C. Biedenharn and H. van Dam, Quantum
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Moshinsky, Marcos (1962). "→igner coefficients for the SU3 group and some applications". Rev. Mod.
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Draayer, J. P.; Akiyama, Yoshimi (1973). "→igner and Racah coefficients for SU3". J. Math. Phys. 14
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Akiyama, Yoshimi; Draayer, J. P. (1973). "A users' guide to fortran programs for →igner and Racah
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Chen, JinQuan; Gao, MeiJuan; Chen, ↓uanGen (1984). "The ClebschGordan coefficient for
SU(m/n) Gel'fand basis". J. Phys. A. 17 (3): 481. Bibcode:1984JPhA...17..727K. doi:10.1088/0305
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Srinivasa Rao, K. (1985). "Special topics in the quantum theory of angular momentum". Pramana. 24
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→ei, 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..222→.
doi:10.1016/S00104655(99)002325.
Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Precalculated →igner 3j, 6j and Gaunt
Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
External links
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3j symbol Wikipedia
Stone, Anthony. "→igner coefficient calculator".
Volya, A. "ClebschGordan, 3j and 6j Coefficient →eb Calculator". (Numerical)
Stevenson, Paul. "ClebschOMatic". Bibcode:2002CoPhC.147..853S. doi:10.1016/S0010
4655(02)004629.
369jsymbol calculator at the Plasma Laboratory of →eizmann Institute of Science (http://plasmagate.
weizmann.ac.il/369j.html) (Numerical)
Frederik J Simons: Matlab software archive, the code THREEJ.M (http://geoweb.princeton.edu/peopl
e/simons/software.html)
Sage (mathematics software) (http://www.sagemath.org/) Gives exact answer for any value of j, m
Johansson, H.T.; Forssén, C. "(→IG↓JPF)". (accurate; C, fortran, python)
Johansson, H.T. "(FAST→IG↓J)". (fast lookup, accurate; C, fortran)
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