Santos, Robenilson F; Bitencourt, Ana Carla P; Ragni, Mirco; Prudente, Frederico V; Coletti, Cecilia; Marzuoli, Annalisa; Aquilanti, Vincenzo
The Wigner 9j symbols of the first kind-also known as Fano X-coefficients-serve to connect different addition schemes of four angular momenta, widely known examples being the LS and the jj couplings in atomic, molecular, and nuclear spectroscopies. Here, we also consider alternative sequences of binary couplings of four angular momenta, which are dealt through the 9j symbols of the second kind, and are explicitly given by the pentagonal (or Biedenharn-Elliott) identity. These coefficients are essential ingredients in the quantum-mechanical treatments of rotational and polarization phenomena in reaction dynamics and photoinduced processes. We also emphasize the combinatorial structure underlying the extended construction of a previously introduced truncated icosahedral "abacus", and provide extensions useful for algebraical manipulations, semiclassical interpretations, and computational applications, including all the 120 addition schemes.
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
We derive an asymptotic formula for the Wigner 12j symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the 12j symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner 9j symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave functions to derive asymptotic formulas for the 9j symbol with small and large angular momenta. When applying the same technique to the 12j symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the 12j symbol is expressed in terms of the vector diagram for a 9j symbol. This illustrates a general geometric connection between asymptotic limits of the various 3nj symbols. This work contributes an asymptotic formula for the 12j symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner 15j symbol with two small angular momenta.
Robinson, Shadow J. Q.; Zamick, Larry
In a previous work [S.J.Q. Robinson and Larry Zamick, Phys. Rev. C 63, 064416 (2001)] we studied the effects of setting all two body T=0 matrix elements to zero in shell model calculations for 43Ti (43Sc) and 44Ti. The results for 44Ti were surprisingly good despite the severity of this approximation. In single-j shell calculations (fn7/2) degeneracies arose between the T=12 I=(12)-1 and (132)-1 states in 43Sc as well as the T=12 I=(132)-2, (172)-1, and (192)-1 in 43Sc. For 44Ti the T=0 states 3+2, 7+2, 9+1, and 10+1 are degenerate as are the 10+2 and 12+1 states. The degeneracies can be explained by certain 6j symbols and 9j symbols either vanishing or being equal as indeed they are. Previously we used Regge symmetries of 6j symbols to explain the vanishing 6j and 9j symbols. In this work a simpler, more physical method is used. This is Talmi's method of calculating coefficients of fractional parentage (cfp) for identical particles to states which are forbidden by the Pauli principle. This is done for both the one particle cfp to handle 6j symbols and the two particle cfp for the 9j symbols. From this we learn that the common thread for the angular momenta I for which the above degeneracies occur is that these angular momenta cannot exist in the calcium isotopes in the f7/2 shell. There are no T=32 f37/2 states with angular momenta 12, 132, 172, and 192. In the same vein there are no T=2 f47/2 states with angular momenta 3, 7, 9, 10, or 12. For these angular momenta, all the states can be classified by the dual quantum numbers (Jπ,Jν).
Anderson, Roger W.; Aquilanti, Vincenzo; da Silva Ferreira, Cristiane
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.
Haggard, Hal Mayi
comparison of this spectrum with that found in loop gravity shows excellent agreement. This provides a simplified derivation of the quantization of space that strengthens earlier proposals along these lines. The second application is an asymptotic formula for the 9j-symbol including its amplitude, phase, and all of the phase adjustments. The 9j-symbol is a more complex spin network than has been treated at this level of detail before and arises as part of the vertex amplitude in spin foams, the loop gravity analog of the path integral approach to quantum gravity. More broadly this quantitative result provides further motivation for developing the asymptotics of higher 3nj-symbols; in the long term these asymptotics, which are accurate even for small quantum numbers, may furnish an effective computational tool for bridging loop gravity predictions to testable experiments.
Zamick, Larry; Robinson, Shadow Jq
In shell model calculations for ^43Ti and ^44Ti not perfect but surprisingly good results are obtained when all the T=0 two body matrix elements are set equal to zero. In this model and in the single j shell approximation (j=f_7/2) many degeneracies arise. For example for the T=1/2 states in ^43Ti(^43Sc) the I=1/2_1^- and 1/32_1^- states are degenerate as are the 1/32_2^-, 1/72_1^- and 1/92_1^- T=1/2 states. In ^44Ti the T=0 states 3^+_2, 7^+_2,9^+_1, and 10^+1 are degenerate and so are the 10^+2 and 12^+1 states. Concerning the 1/2_1^- and 1/32_1^- we find that both have (J_p,J_n) configuration (4,7/2). For the 3^+_2, 7^+_2,9^+_1, and 10^+1 all four states have the configurations (4,6) and (6,4). This means that couplings to other states will vanish. This means that certain 6j and 9j symbols will vanish e.g. j & j& 4 j&1/32& 6 \\=0 and j & j& 6 j&j& 6 4 & 6 &10 \\=0. One can explain these vanishings in terms of Talmi's method of calculating coeffiecients of fractional parentage to states not allowed by the Pauli principle. For example for the T=3/2 states of ^43Ca there are no f_7/2^3 I=1/32^- states. Hence the c.f.p. to these states must vanish. One c.f.p. contains the above 6j symbol and so this 6j symbol will vanish. For the T=2 states in ^44Ca there is no f_7/2^4 state with I=10^+. One of the two particle c.f.p to this state is proportional to the above 9j symbol and so the 9j must vanish. Note that we are using arguments about T=3/2 states to explain degeneracies of T=1/2 states; and we are using arguments about T=2 states to explain degeneracies of T=0 states. When T=0 two body matrix elements are reintroduced the 9^+1 and 10^+1 are no longer degenerate and the splitting even in a full fp space calculation is due almost entirely to the T=0 matrix elements. The common thread for the T=1/2 and T=0 states that are degenerate is that they have angular momentum which in the single j shell calculation cannot occur for identical particles. For these angular momenta
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.