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A new property of determinants and the generating function of Wigner 9-j symbols

Mehdi Hage-Hassan

Published May 30, 2020• 0 citations
OpenAlex

Abstract

We find a new property of determinants very useful for the computation of the generating function of the recouplings coefficients of the angular momentum. Using the generating function method, we simply transform the treatment of Schwinger's couplings in the bosons basis to the analytic space (Fock space). We find the exact generating function of the 9-j coefficients using the property of the determinants and the Gaussian integral. We also give a simple proof of the celebrated Mehler's formula.

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We find a new property of determinants very useful for the computation of the generating function of the recouplings coefficients of the angular momentum. Using the generating function method, we simply transform the treatment of Schwinger's couplings in the bosons basis to the analytic space (Fock space). We find the exact generating function of the 9-j coefficients using the property of the determinants and the Gaussian integral. We also give a simple proof of the celebrated Mehler's formula.
We find a new property of determinants very useful for the computation of the generating function of the recouplings coefficients of the angular momentum. Using the generating function method, we simply transform the treatment of Schwinger's couplings in the bosons basis to the analytic space (Fock space). We find the exact generating function of the 9-j coefficients using the property of the determinants and the Gaussian integral. We also give a simple proof of the celebrated Mehler's formula.
We analyze the asymptotics of the Wigner 3j3j3j-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 Bohr-Sommerfeld 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.
We present a novel method for deriving some new quantum operator formulas by carrying out the integrations within a normal product symbol. The idea underlying this method is combining the properties of normal product and of coherent state together. For example, we can deduce operator ordering theorems, the Fujiwara formula, and some disentangling theorems simply and directly. By means of this method we also get the explicit operator form and coherent state form for the Wigner function with these new forms, the wigner function's eigenstate is found and its zero point value property is discussed. Furthermore some famous quantum unitary transformations and the transformed matrix element for functional obtained conveniently. integral may be also
The angular momentum states may be realized by the wave functions of two dimensional harmonic oscillators in polar coordinates. Using the generating function of these wave functions, Schwinger's generating function of Clebsh-Gordan in Fock space and Gauss integral, we determine the generating function of the couplings function of two angular momentums in coordinates representations. We deduce from the last function the connection of the wave function of the hydrogen atom and the four dimensional harmonic oscillator. We also find the expression of the relation between the spherical and parabolic wave functions of hydrogen atom. In addition, we find a new bilinear generating function involving the products of Laguerre polynomials.