Asymptotics of Clebsch–Gordan coefficients

In this work we present the details of calculations we previously performed for the large j behavior of certain 3j and 9j symbols.In this paper we focus on equations (11 and 13), and (23 and 24) of the work of Kleszyk and Zamick [1]. In particular we consider the case when the total angular momentum I is equal to I max − 2n and I max ≡ 4j − 2, and n = 0, 1, 2, ... We take the limit of large j where n becomes much smaller than j. For convenience, we also define J = 2j, where j is the total angular momentum of a single particle.We first address the 3j coefficient, using the formula Eq. (13) We express the total angular momentum I using a new variable m such that I = 4j − 2m, where this time m = 1, 2, 3, .... We can separate parts of the 3j which now becomeswhere the 6 factors N i are:We use the Stirling approximation,and it should be noted that the approximation approaches the true value asymptotically. Now we can write: N i = (α i + β i m + γ i J) with differing constant coefficients. In Eq.(2a) we give the contribuition of −N, ln √ 2πN , α ln N, mβ ln N, and γJ ln N . For the latter we break things up into (a) "extreme" and (b) "next order". This is necessary because "next order" has contributions comparable to those in "−N ". (6) 1 arXiv:1406.4495v4 [nucl-th]

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“…We note other work on asymptotics of CG coefficients by Reinsch and Morehead [3]. In their work they define…”

Section: 143mentioning

confidence: 99%

Asymptotics of the3jand9jcoefficients

2014

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“…We note other work on asymptotics of CG coefficients by Reinsch and Morehead [12]. In their work they define β = ((j 1 + j 2 − j)(j + j 2 − j 1 )(j + j 1 − j 2 )(j 1 + j 2 + j)) j + j 1 + j 2 j + j 1 + j 2 + 1…”

Section: A1 the Unitary 9j Coefficientmentioning

confidence: 99%

Analytical and numerical calculations for the asymptotic behavior of unitary9jcoefficients

Kleszyk

Zamick

2014

Phys. Rev. C

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“…It is not obvious to us what the Langer modification has to do with the 1/2 that occurs in the present context, nor are we aware of any general rules about when in the asymptotics of angular momentum theory it is correct to replace a classical j by j +1/2 (instead of [j(j +1)] 1/2 or something else). Schulten and Gordon (1975b) and Reinsch and Morehead (1999) obtain the 1/2 as a part of their proper semiclassical analyses. Biedenharn and Louck (1981b) also speculate on the significance of the 1/2.…”

Section: Quantizing the Jm-torimentioning

confidence: 99%

“…Somewhat later Biedenharn and Louck (1981b) presented a review and commentary of the results of Ponzano and Regge, as well as a proof based on showing that the result satisfies asymptotically a set of defining relations for the 6j-symbol. More recently the asymptotics of the 3j-symbol was derived again by Reinsch and Morehead (1999), working with an integral representation constructed out of Wigner's singleindex sum for the Clebsch-Gordan coefficients. About the same time, Roberts (1999) derived the Ponzano and Regge results for the 6j-symbol, using methods of geometric quantization.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical analysis of Wigner 3j-symbol

Aquilanti

Haggard

Littlejohn

et al. 2007

J. Phys. A: Math. Theor.

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Abstract.We analyze the asymptotics of the Wigner 3j-symbol as a matrix element connecting eigenfunctions of a pair of integrable systems, obtained by lifting the problem of the addition of angular momenta into the space of Schwinger's oscillators. A novel element is the appearance of compact Lagrangian manifolds that are not tori, due to the fact that the observables defining the quantum states are noncommuting. These manifolds can be quantized by generalized BohrSommerfeld rules and yield all the correct quantum numbers. The geometry of the classical angular momentum vectors emerges in a clear manner. Efficient methods for computing amplitude determinants in terms of Poisson brackets are developed and illustrated.

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Asymptotics of Clebsch–Gordan coefficients

Cited by 18 publications

References 6 publications

Asymptotics of the3jand9jcoefficients

Asymptotics of the3jand9jcoefficients

Analytical and numerical calculations for the asymptotic behavior of unitary9jcoefficients

Semiclassical analysis of Wigner 3j-symbol

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