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The Journal of Chemical Physics

Global uniform semiclassical approximation for Clebsch-Gordan coefficients

Hamutal Engel, Kenneth G. Kay

Published Mar 4, 2008• 7 citations
Publisher

Abstract

Semiclassical integral representations, analogous to initial value expressions for the propagator, are presented for the Clebsch-Gordan angular momentum coupling coefficients. Two forms (L and R types) of the approximation are presented. For each form, new non-Gaussian expressions, which are specifically adapted to the nature of angular momentum variables, are proposed in place of the familiar Gaussian coherent state functions. With these non-Gaussian kernels, it is found that the present treatments are capable of accuracy similar to that obtained from a uniform Airy approximation. Although the present semiclassical approximations involve only real-valued angle variables, associated with sets of angular momenta that are related by ordinary, real, classical transformations, the treatments produce accurate results not only for classically allowed choices of quantum numbers but also for very strongly classically forbidden values.

DOI
10.1063/1.2838981

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Semiclassical integral representations, analogous to initial value expressions for the propagator, are presented for the Clebsch-Gordan angular momentum coupling coefficients. Two forms (L and R types) of the approximation are presented. For each form, new non-Gaussian expressions, which are specifically adapted to the nature of angular momentum variables, are proposed in place of the familiar Gaussian coherent state functions. With these non-Gaussian kernels, it is found that the present treatments are capable of accuracy similar to that obtained from a uniform Airy approximation. Although the present semiclassical approximations involve only real-valued angle variables, associated with sets of angular momenta that are related by ordinary, real, classical transformations, the treatments produce accurate results not only for classically allowed choices of quantum numbers but also for very strongly classically forbidden values.
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.
A semiclassical expression for matrix elements of an arbitrary operator with respect to the eigenstates of an integrable Hamiltonian is derived. This is essentially the Heisenberg correspondence principle, and it is shown via the Weyl correspondence that the approximation is valid through the lowest two orders in \ensuremath{\Elzxh}. The result is used to prove that an asymptotic form of the Clebsch-Gordan coefficients for two large and one small angular momenta is valid through two orders. \textcopyright{} 1996 The American Physical Society.
The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.
The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.