A unified approach for exact calculation of angular momentum coupling and recoupling coefficients

Abstract

In this new version of the program, I have unified the computation and tabulation of all the recoupling coefficients or 3n−j(n≥2) symbols for all the ranges of quantum angular momentum arguments based on the reduced 6−j symbols of the same type that I have defined. Additionally, I have added one specific subroutine for the computation and tabulation of the 12−j symbol of the first kind at this time. Moreover, I have made the module for the previous codes that transforms the tables to the strictly exact decimal values. New version program summaryProgram Title: 369j.Catalogue identifier: ADKL_v2_0.Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADKL_v2_0.html.Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland.Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html.No. of lines in distributed program including test data, etc.: 1742.No. of bytes in distributed program, including test data etc.: 9631.Distribution format: tar.gz.Programming language: Fortran.Computer: Any.Operating system: Any.Keywords: reduced 6−j symbols, 12−j symbols, 3n−j symbols, tabulation, decimal value, strict exactness.Classification: 4.1.Catalogue identifier of previous version: ADKL_v1_0.Journal reference of previous version: See reference [1].Does the new version supersede the previous version?: Yes.Nature of problem: Quantum angular momentum coupling and recoupling coefficients occur in almost all of the areas grounded with quantum mechanics. The strict exact, direct, or most efficient calculation or tabulation of all these coefficients for all the ranges of quantum angular momentum arguments are required in a variety of situations from subatomic physics to astrophysics.Solution method: I have invented the algebraic formulas for the 6−j symbol in terms of the reduced 6−j symbols and for all the other 3n−j(n≥3) symbols in terms of the summation over the products of the reduced 6−j symbols. I have devised the combinatorial expressions for the reduced 6−j symbols in terms of the summation over the products of the binomial coefficients, similar to the case of the 3−j symbol. I have created the recursive formulas for the binomial coefficients for their calculation instead of the evaluation of the factorials of integers. I have also developed and synthesized two different number representations in all the computations accordingly. They are the prime number representation for the prefactor and the 32768-base number representation for the summation terms [1–3].Reasons for the new version: I want to keep track of the advancements in my formulation for the recoupling coefficients, and therefore to update the applications in my computation and tabulation of these coefficients at the ways stated [2, 3].Summary of revisions: (1) I have revised the program with the reduced 6−j symbols of the same type that I have invented for the unifying computation and tabulation of all the recoupling coefficients or 3n−j(n≥2) symbols. (2) I have added the specific subroutine to compute and tabulate the 12−j symbol of the first kind at this time. (3) I have made the module for the previous codes to transform the tables into the strictly exact decimal values.11I again acknowledge one of the referees for my paper [1] for writing the codes of this module.Running time: The time taken for calculating 3−j, 6−j, 9−j symbols, and 12−j symbols of the first kind for small angular momenta is instantaneous. It is still less than 0.5 s when quantum angular momentum arguments are very large.