Exact Computation and Asymptotic Approximations of 6j Symbols: Illustration of Their Semiclassical Limits

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). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem 110: 731-742, 2010

[1]  Klaus Schulten,et al.  Exact recursive evaluation of 3j- and 6j-coefficients for quantum- mechanical coupling of angular momenta , 1975 .

[2]  J. Labarthe.  Generating functions for the coupling-recoupling coefficients of SU(2) , 1975 .

[3]  V. B. Uvarov,et al.  Classical Orthogonal Polynomials of a Discrete Variable , 1991 .

[4]  L. Freidel,et al.  Asymptotics of 6j and 10j symbols , 2002, hep-th/0209134.

[5]  Vincenzo Aquilanti,et al.  Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem , 2001 .

[6]  Donald E. Neville,et al.  A Technique for Solving Recurrence Relations Approximately and Its Application to the 3‐J and 6‐J Symbols , 1971 .

[7]  Gediminas Gaigalas,et al.  Maple procedures for the coupling of angular momenta. VIII. Spin-angular coefficients for single-shell configurations , 2005, Comput. Phys. Commun..

[8]  Vincenzo Aquilanti,et al.  Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm , 2000 .

[9]  M. Reid,et al.  Geometry and Topology , 2005, Mathematical Constants II.

[10]  Vincenzo Aquilanti,et al.  3nj-symbols and harmonic superposition coefficients: an icosahedral abacus , 2001 .

[11]  Klaus Schulten,et al.  Semiclassical approximations to 3j- and 6j-coefficients for quantum­ mechanical coupling of angular momenta An inverse problem in statistical mechanics Direct determination of the Iwasawa decomposition for noncompact , 1975 .

[12]  Vincenzo Aquilanti,et al.  Orthogonal polynomials of a discrete variable as expansion basis sets in quantum mechanics: Hyperquantization algorithm , 2003 .

[13]  Vincenzo Aquilanti,et al.  ANGULAR AND HYPERANGULAR MOMENTUM COUPLING COEFFICIENTS AS HAHN POLYNOMIALS , 1995 .

[14]  Shan-Tao Lai,et al.  CALCULATION OF 3N-J SYMBOLS BY LABARTHE'S METHOD , 1997 .

[15]  Rene F. Swarttouw,et al.  The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[16]  T. Inghoff,et al.  Maple procedures for the coupling of angular momenta. V. Recoupling coefficients ? ? This program ca , 2001 .

[17]  Vincenzo Aquilanti,et al.  Angular and hyperangular momentum recoupling, harmonic superposition and Racah polynomials: a recursive algorithm , 2001 .

[18]  S. Fritzsche,et al.  Maple procedures for the coupling of angular momenta. VII. Extended and accelerated computations , 2003 .

[19]  Justin Roberts.  Classical 6j-symbols and the tetrahedron , 1998 .

[20]  Liqiang Wei.  Corrigendum: Universal factorization of 3n − j(j > 2) symbols of the first and second kinds for SU(2) group and their direct and exact calculation and tabulation , 2003, math-ph/0306040.

[21]  Asymptotics of Clebsch–Gordan coefficients , 1999, math-ph/9906007.

[22]  A. Deveikis,et al.  The analytical Scheme calculator for angular momentum coupling and recoupling coefficients , 2005, Comput. Phys. Commun..

[23]  Vincenzo Aquilanti,et al.  Exact computation and large angular momentum asymptotics of 3nj symbols: Semiclassical disentangling of spin networks. , 2008, The Journal of chemical physics.

[24]  Vincenzo Aquilanti,et al.  The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors. , 2006, The Journal of chemical physics.

[25]  Klaus Schulten,et al.  Recursive evaluation of 3j and 6j coefficients , 1984 .

[26]  A Macsyma program for computing analytic Clebsch–Gordan coefficients of U(3)⊃SO(3) , 2001 .

[27]  A study of the helium 31D→21P→11S cascade with algebraic computing , 1993 .

[28]  Vincenzo Aquilanti,et al.  Semiclassical analysis of Wigner 3j-symbol , 2007, quant-ph/0703104.

[29]  G. Gaigalas,et al.  Maple procedures for the coupling of angular momenta. VI. LS–jj transformations☆ , 2002 .

[30]  D. Varshalovich,et al.  Quantum Theory of Angular Momentum , 1988 .

[31]  R. Askey.  Duality for classical orthogonal polynomials , 2005 .

[32]  Unified approach for exact calculation of angular momentum coupling and recoupling coefficients , 1999 .

[33]  Vincenzo Aquilanti,et al.  Hydrogenic orbitals in Momentum space and hyperspherical harmonics: Elliptic Sturmian basis sets , 2003 .

[34]  Igal Talmi,et al.  Spectroscopic and Group Theoretical Methods in Physics , 1970 .

[35]  V. Aquilanti,et al.  Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity , 2008 .