Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

Mehdi Hage-Hassan

Published May 28, 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.

Related papers

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

Mehdi Hage-Hassan

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.

•May 30, 2020•0 citations

Normal Product Expansion of Rotation Operators Gained by Coherent State Method

Fan Hang-yi

In Schwinger boson representation of angular momentum theory we derive new normal product expansion of rotation operators which can generate new operator identities in the Hilbert space to which SU 2 group maps, some properties and applications of them are presented.

Communications in Theoretical Physics•Mar 1, 1987•6 citations

Matrix elements of off-diagonal operator equivalents

Wolfgang Witschel

A simple derivation of off-diagonal operator equivalents and their matrix elements is given in the framework of Schwinger's coupled boson representation of angular momentum theory. The operator equivalents are up to a factor identical with those recently given by Atkins and Seymour, but they are derived differently without making use of properties of spherical harmonics, symmetrization, 3j coefficients and the Wigner-Eckart theorem.

Molecular Physics•Aug 1, 1973•7 citations

Angular momentum, Hydrogen-oscillator connection and the relation between hydrogen atom basis

Mehdi Hage-Hassan

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.

•Apr 9, 2016•0 citations

Eigenvalues and Eigenvectors of Angular Momentum Operator Jx without the Theory of Rotations

Lorenzo M. Narducci, Miguel Orszag

We propose an algebraic diagonalization procedure for the angular momentum matrix Jx which is based on the introduction of a suitable generating function. The consideration of the analytic properties of the generating function leads to the eigenvalues of the problem. The coefficients of the power series expansion of this function generate the components of the eigenvectors of Jx. This procedure provides the beginning student of quantum mechanics with an alternative derivation of the eigenvalue spectrum of the angular momentum.

American Journal of Physics•Dec 1, 1972•19 citations

Generating function method and its applications to Quantum, Nuclear and the Classical Groups

Mehdi Hage-Hassan

The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear and group theory textbooks. We find simply: the generating function of the harmonic oscillator, the Feynman propagators of the oscillator and the oscillator in uniform magnetic field. We derive the invariants of SU(2) and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitz quadratic transformations. We show that the cross-product exist only in E3 and E7. We determine the {p} representation of hydrogen atom in three and n-dimensions. We generalize the Cramer's rule for the calculation of the rotational spectrum of the nucleus. We find the expression of the Hamiltonian in terms of quasi-bosons for study the collective vibration. We determine the basis and the expressions of 3-j symbols of SU (3) and SU(n).We find the Schrödinger equation from Hamilton-Jacobi formalism. We present these applications in independent chapters.

arXiv:1203.2892•Mar 13, 2012•0 citations
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.
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.
We propose an algebraic diagonalization procedure for the angular momentum matrix Jx which is based on the introduction of a suitable generating function. The consideration of the analytic properties of the generating function leads to the eigenvalues of the problem. The coefficients of the power series expansion of this function generate the components of the eigenvectors of Jx. This procedure provides the beginning student of quantum mechanics with an alternative derivation of the eigenvalue spectrum of the angular momentum.
The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear and group theory textbooks. We find simply: the generating function of the harmonic oscillator, the Feynman propagators of the oscillator and the oscillator in uniform magnetic field. We derive the invariants of SU(2) and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitz quadratic transformations. We show that the cross-product exist only in E3 and E7. We determine the {p} representation of hydrogen atom in three and n-dimensions. We generalize the Cramer's rule for the calculation of the rotational spectrum of the nucleus. We find the expression of the Hamiltonian in terms of quasi-bosons for study the collective vibration. We determine the basis and the expressions of 3-j symbols of SU (3) and SU(n).We find the Schrödinger equation from Hamilton-Jacobi formalism. We present these applications in independent chapters.