Hertz-Kintish, Daniel; Zamick, Larry; Kleszyk, Brian
We investigate the large j behavior of certain 3 j and 9 j symbols, where j is the total angular momentum of one particle in a given shell. Our motivation is the problem of maximum J pairing in nuclei, along with the more familiar J = 0 pairing. Maximum J pairing leads to an increase in J = 2 coupling of two protons and two neutrons relative to J = 0 . We find that a coupling unitary 9 j symbol (U 9 j) is very weak as j increases, leading to wavefunctions which are to an excellent approximation single U 9 j coefficients. Our study of the large j behavior of coupling unitary 9 j symbols is through the consideration of the case when the total angular momentum I is equal to Imax - 2 n and Imax ≡ 4 j - 2 , where n = 0 , 1 , 2 , ... . We here derive asymptotic approximations of coupling 3 j symbols and find that the 3 j ~j - 3 / 4 in the high j limit. One major analytical tool we used is the Stirling Approximation. Through analytical, numerical, and graphical methods, we show the power law behavior of the coupling unitary 9 j symbols in the n / j << 1 limit, i.e. U 9 j ~j-n . Power-law behavior is evident if there is a linear dependence of ln | U 9 j | vs. ln j . We also present some examples of percent errors in our approximations. We investigate the large j behavior of certain 3 j and 9 j symbols, where j is the total angular momentum of one particle in a given shell. Our motivation is the problem of maximum J pairing in nuclei, along with the more familiar J = 0 pairing. Maximum J pairing leads to an increase in J = 2 coupling of two protons and two neutrons relative to J = 0 . We find that a coupling unitary 9 j symbol (U 9 j) is very weak as j increases, leading to wavefunctions which are to an excellent approximation single U 9 j coefficients. Our study of the large j behavior of coupling unitary 9 j symbols is through the consideration of the case when the total angular momentum I is equal to Imax - 2 n and Imax ≡ 4 j - 2 , where n = 0 , 1 , 2 , ... . We here
Anderson, Roger W.; Aquilanti, Vincenzo; Silva Ferreira, Cristiane da
Spin networks, namely, the 3nj symbols of quantum angular momentum theory and their generalizations to groups other than SU(2) and to quantum groups, permeate many areas of pure and applied science. The issues of their computation and characterization for large values of their entries are a challenge for diverse fields, such as spectroscopy and quantum chemistry, molecular and condensed matter physics, quantum computing, and the geometry of space time. Here we record progress both in their efficient calculation and in the study of the large j asymptotics. For the 9j symbol, a prototypical entangled network, we present and extensively check numerically formulas that illustrate the passage to the semiclassical limit, manifesting both the occurrence of disentangling and the discrete-continuum transition.
Deveikis, A.; Kuznecovas, A.
We describe a Scheme implementation of the interactive environment to calculate analytically the Clebsch-Gordan coefficients, Wigner 6 j and 9 j symbols, and general recoupling coefficients that are used in the quantum theory of angular momentum. The orthogonality conditions for considered coefficients are implemented. The program provides a fast and exact calculation of the coefficients for large values of quantum angular momenta. Program summaryTitle of program:Scheme2Clebsch Catalogue number:ADWC Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWC Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed:Any Scheme-capable platform Operating systems under which the program has been tested: Windows 2000 Programming language used:Scheme Memory required to execute with typical data:50 MB (≈ size of DrScheme, version 204) No. of lines in distributed program, including test data, etc.: 2872 No. of bytes in distributed program, including test data, etc.: 109 396 Distribution format:tar.gz Nature of physical problem:The accurate and fast calculation of the angular momentum coupling and recoupling coefficients is required in various branches of quantum many-particle physics. The presented code provides a fast and exact calculation of the angular momentum coupling and recoupling coefficients for large values of quantum angular momenta and is based on the GNU Library General Public License PLT software http://www.plt-scheme.org/. Method of solution:A direct evaluation of sum formulas. A general angular momentum recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta is expressed as a sum over products of the Clebsch-Gordan coefficients. Restrictions on the complexity of the problem:Limited only by the DrScheme implementation used to run the program. No limitation inherent in the code. Typical running time:The Clebsch
Zamick, Larry; Mekjian, Aram
We use the short notation for a unitary 9j symbol U9j(Ja,Jb)=<(jj)Ja(jj)Ja|(jj)Jb(jj)Jb>I=0 The wave fcn of a state of 44Ti with ang momentum I can be written as sum D(Jp,Jn) [(jj)Jp (jj)Jn]I. For the I=0 ground stae Jp=Jn. We found a new relationship SumJp U9j(Jp,Jx) D(Jp,Jp)= 1/2 D(Jx,Jx) for T=0 and =-D(Jx,Jx) for T=2. We could explain this by regarding U9j for even Jp,Jx as a square matrix hamiltonian, which, when diagonalized has eigenvalues of 1/2(triply degenerate) and -1(singly degenerate) corresponding to T=0 and T=2 respectively.*This theorem is useful,in the context of 2 nucleon transfer, for counting the number of pairs of particles in 44Ti with even Jx.The expressions simplifies to 3|D(Jx,Jx|^2,thus eliminating a complex 9jsymbol A deeper understanding of this result arises if we consider the strange interplay of angular momentum and isospin. Consider the interaction 1/4-t(1).t(2),which is unity for T=0 states and zero for T=1. For n nucleons with isospin T the eigenvalues are n^2/8+n/4-T(T+1)/2 But if we evaluate this with the usual Racah algebra then we note that in the single j shell the interaction can also be written as <(jj)Ia V (jj)Ia>= (1-(-1)^Ia)/2 i.e. the interaction acts only in odd J states since they have isospin T=0.In 44Ti the matrix element of the hamiltonian is [2+2U9j(Jp,Jx)].Connecting this with the isospin expression gives us the eigenvalues above for U9j. * L.Zamick, E. Moya de Guerra,P.Sarriguren,A.Raduta and A. Escuderos, preprint.