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Physical Review A

New quantum numbers in collision theory. II. Angular momentum diagrams and interpretation

Chun‐Woo Lee

Published Aug 1, 1986• 9 citations
Publisher

Abstract

Geometrical relationships illustrate the significance of the transformation between new and orbital quantum numbers. The semiclassical analysis of 6j coefficients by Ponzano and Regge clarifies the geometrical relationships and identifies a classically forbidden range, which can be singled out for its small contribution to the summation over the partial waves relevant to observable parameters (differential cross section and orientation).

DOI
10.1103/physreva.34.959

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Asymptotic Limits of the Wigner $12J$-Symbol in Terms of the Ponzano-Regge Phases

Yu, Liang

There are two types of asymptotic formulas for the $12j$ symbol with one small and 11 large angular momenta. We have derived the first type of formula previously in [L. Yu, Phys. Rev. A84 022101 (2011)]. We will derive the second type in this paper. We find that this second asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram associated with two $6j$ symbols, namely, the vector diagram of two adjacent tetrahedra sharing a common face. As a result, two sets of Ponzano-Regge phases appear in the asymptotic formula. This work contributes another asymptotic formula of the Wigner $12j$ symbol to the re-coupling theory of angular momenta.

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Geometrical relationships illustrate the significance of the transformation between new and orbital quantum numbers. The semiclassical analysis of 6j coefficients by Ponzano and Regge clarifies the geometrical relationships and identifies a classically forbidden range, which can be singled out for its small contribution to the summation over the partial waves relevant to observable parameters (differential cross section and orientation).
Scattering geometries are represented by tensorial parameters classified according to symmetry under permutations P and Q: P changes a base set of wave functions into its Hermitian conjugate and Q interchanges the directions of incidence and scattering. These parameters interlink initial and final states of projectile and target, a linkage mediated by the angular momentum jt{\mathrm{j}}_{t}jt​ transferred in the collision. Collision cross sections and expectation values of the observables are constructed as sums of geometrical parameters with different symmetries, weighted by dynamical parameters.
The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.
Full-dimensional quantum scattering calculations are reported for ro-vibrational transitions in HD+HD collisions using a highly accurate interaction potential for the H2_22​-H2_22​ system. Several near-resonant ro-vibrational transitions are identified that conserve the overall rotational angular momentum and nearly conserve the internal energy of the collision partners. Key anisotropic terms that drive the rotational transitions and angular momentum partial waves that contribute to low energy resonant features in the energy dependence of the cross sections are identified. The computed results are in agreement with total cross sections reported in previous experimental results, including resonant features in the energy dependence of the cross section. In particular, low-energy cross sections show a strong resonant feature associated with an l=3l=3l=3 partial wave in the incident channel. Rate coefficients for several inelastic rotational and ro-vibrational transitions are reported for temperatures ranging from 0.10.10.1 K to 200200200 K and they display a maximum between 111 K-101010 K reflecting the important contributions from the l=3l=3l=3 shape resonance that occurs around 2.5 K.
Asymptotic Limits of the Wigner 12J12J12J-Symbol in Terms of the Ponzano-Regge Phases
There are two types of asymptotic formulas for the 12j12j12j symbol with one small and 11 large angular momenta. We have derived the first type of formula previously in [L. Yu, Phys. Rev. A84 022101 (2011)]. We will derive the second type in this paper. We find that this second asymptotic formula for the 12j12j12j symbol is expressed in terms of the vector diagram associated with two 6j6j6j symbols, namely, the vector diagram of two adjacent tetrahedra sharing a common face. As a result, two sets of Ponzano-Regge phases appear in the asymptotic formula. This work contributes another asymptotic formula of the Wigner 12j12j12j symbol to the re-coupling theory of angular momenta.