Abstract
This article presents and discusses in detail the results of extensive exact calculations of the most basic ingredients of spin networks, the Racah coefficients ( or Wigner 6j symbols), exhibiting their salient features when considered as a function of two variables - a natural choice due to their origin as elements of a square orthogonal matrix - and illustrated by use of a projection on a square “scree” introduced recently. On these screens, shown are images which provide a systematic classification of features previously introduced to represent the caustic and ridge curves ( which delimit the boundaries between oscillatory and evanescent behaviour according to the asymptotic analysis of semiclassical approaches). Particular relevance is given to the surprising role of the intriguing symmetries discovered long ago by Regge and recently revisited; from their use, together with other newly discovered properties and in conjunction with the traditional combinatorial ones, a picture emerges of the amplitudes and phases of these discrete wavefunctions, of interest in wide areas as building blocks of basic and applied quantum mechanics.
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Aquilanti, V., Bitencourt, A., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity. Physica Scripta 78, 058103 (2008)
Aquilanti, V., Bitencourt, A., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications. Theor. Chem. Accounts 123, 237 (2009)
Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Bloch, F., et al. (eds.) Spectroscopic and Group Theoretical Methods in Physics, pp. 1–58. North–Holland, Amsterdam (1968)
Neville, D.: A technique for solving recurrence relations approximately and its application to the 3 − j and 6 − j symbols. J. Math. Phys. 12, 2438 (1971)
Schulten, K., Gordon, R.: Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta. J. Math. Phys. 16, 1971–1988 (1975)
Schulten, K., Gordon, R.: Exact recursive evaluation of 3j- and 6j-coefficients for quantum mechanical coupling of angular momenta. J. Math. Phys. 16, 1961–1970 (1975)
Ragni, M., Bitencourt, A.C., Aquilanti, V., Anderson, R.W., Littlejohn, R.G.: Exact computation and asymptotic approximations of 6j symbols: Illustration of their semiclassical limits. Int. J. Quantum Chem. 110(3), 731–742 (2010)
Aquilanti, V., Cavalli, S., Coletti, C.: Angular and hyperangular momentum recoupling, harmonic superposition and Racah polynomials. a recursive algorithm. Chem Phys. Letters 344, 587–600 (2001)
Littlejohn, R.G., Yu, L.: Uniform semiclassical approximation for the Wigner 6j-symbol in terms of rotation matrices. J. Phys. Chem. A 113, 14904–14922 (2009)
Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevangee, N., Littlejohn, R., Yu, L.: Semiclassical mechanics of the Wigner 6j-symbol. J. Phys. A 45(065209) (2012), arXiv:1009.2811v2 [math-ph]
Aquilanti, V., Capecchi, G.: Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Accounts 104, 183–188 (2000)
De Fazio, D., Cavalli, S., Aquilanti, V.: Orthogonal polynomials of a discrete variable as expansion basis sets in quantum mechanics. The hyperquantization algorithm. Int. J. Quantum Chem. 93, 91–111 (2003)
Aquilanti, V., Cavalli, S., De Fazio, D.: Angular and Hyperangular Momentum Coupling Coefficients as Hahn Polynomials. J. Phys. Chem. 99(42), 15694–15698 (1995)
Koekoek, R., Lesky, P., Swarttouw, R.: Hypergeometric orthogonal polynomials and their q-analogues. Springer (2010)
Bitencourt, A.C.P., Marzuoli, A., Ragni, M., Anderson, R.W., Aquilanti, V.: Exact and asymptotic computations of elementary spin networks: Classification of the quantum–classical boundaries. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part I. LNCS, vol. 7333, pp. 723–737. Springer, Heidelberg (2012), See arXiv:1211.4993[math-ph]
Varshalovich, D., Moskalev, A., Khersonskii, V.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)
Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Yu, L.: Semiclassical analysis of Wigner 3 j -symbol. J. Phys. A 40(21), 5637–5674 (2007)
Anderson, R.W., Aquilanti, V.: The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors. J. Chem. Phys. 124, 214104 (9 pages) (2006)
Anderson, R.W., Aquilanti, V., da Silva Ferreira, C.: Exact computation and large angular momentum asymptotics of 3nj symbols: semiclassical disentangling of spin networks. J. Chem. Phys. 129, 161101–161105 (2008)
Anderson, R.W., Aquilanti, V., Bitencourt, A.C.P., Marinelli, D., Ragni, M.: The screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and hamiltonian dynamics. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Quang, N.H., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013, Part II. LNCS, vol. 7972, pp. 46–59. Springer, Heidelberg (2013)
Aquilanti, V., Marinelli, D., Marzuoli, A.: Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials. J. Phys. A: Math. Theor. 46, 175303 (2013), arXiv:1301.1949v2 [math-ph]
Gilmore, R.: Catastrophe Theory for Scientists and Engineers. Dover, New York (1993)
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Ragni, M., Littlejohn, R.G., Bitencourt, A.C.P., Aquilanti, V., Anderson, R.W. (2013). The Screen Representation of Spin Networks: Images of 6j Symbols and Semiclassical Features. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2013. ICCSA 2013. Lecture Notes in Computer Science, vol 7972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39643-4_5
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DOI: https://doi.org/10.1007/978-3-642-39643-4_5
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