Searcharxiv⌕ Search

arXiv subjects

Mehdi Hage-Hassan

Publications and source records attributed to Mehdi Hage-Hassan.

12 recordsLinked to original sources
arXiv2014

A note on the normalization of the momentum eigenfunctions and Dirac delta function

We determine the generating function of the harmonic oscillator by a new method. Using this generating function we derive the eigenfunctions of the moment p. We find that the normalization of these eigenfunctions is a real and not complex number with phase factor chosen equal one (standard books of quantum mechanics). We prove that the integral of the delta function is equal to one and we derive the oscillator propagator.

quant-ph↗
arXiv2013

Applications of generating function method to the symmetry and the Kronecker products of SU(n) representations

Using the generating function of SU(n) we find the conjugate state of SU(n) basis and we find in terms of Gel'fand basis of SU(3(n-1)) the representation of the invariants of the Kronecker products of SU(n). We find a formula for the number of the elementary invariants of SU(n). We apply our method to the coupling of SU(3) and we find a new expression of the isoscalar of Wigner symbols ($λ10,λ2 μ2; λ3 μ3$).

math-ph↗
arXiv2013

A note on Quarks and numbers theory

We express the basis vectors of Cartan fundamental representations of unitary groups by binary numbers. We determine the expression of Gel'fand basis of SU (3) based on the usual subatomic quarks notations and we represent it by binary numbers. By analogy with the mesons and quarks we find a new property of prime numbers.

physics.gen-ph↗
arXiv2012

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

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.

math-ph↗
arXiv2011

The "boson polynomials" of Gel'fand basis and the analytic expression of Wigner's coefficients with multiplicity for the canonical basis of unitary groups

In this paper we present the generating function method for the derivation of bosons polynomials of Gel'fand basis and Wigner coefficients for the canonical basis of SU(n). We find a new analytic polynomial basis of SU(4) with the exact number of summations, five only. We find also a new algebraic expression of Wigner coefficient with multiplicity for the canonical basis and the isoscalors factors of SU (3) with only. three summations.

math-ph↗
arXiv2010

On Fock-Bargmann space, Dirac delta function, Feynman propagator, angular momentum and SU(3) multiplicity free

The Dirac delta function and the Feynman propagator of the harmonic oscillator are found by a simple calculation using Fock Bargmann space and the generating function of the harmonic oscillator. With help of the Schwinger generating function of Wigner's D-matrix elements we derive the generating function of spherical harmonics, the quadratic transformations and the generating functions of: the characters of SU (2), Legendre and Gegenbauer polynomials. We also deduce the van der Wearden invariant of 3-j symbols of SU (2). Using the Fock Bargmann space and its complex conjugate we find the integral representations of 3j symbols, function of the series, and from the properties of we deduce a set of generalized hypergeometric functions of SU (2) and from Euler's identity we find Regge symmetry. We find also the integral representation of the 6j symbols. We find the generating function and a new expression of the 3j symbols for SU (3) multiplicity free. Our formula of SU (3) is a product of a constant, 3j symbols of SU (2) by . The calculations in this work require only the Gauss integral, well known to undergraduates.

math-ph↗
arXiv2009

Generalization of Cramer's rule and its application to the projection of Hartree-Fock wave function

We generalize the Cramer's rule of linear algebra. We apply it to calculate the spectra of nucleus by applying Hill-Wheeler projection operator to Hartree-Fock wave function, and to derive Löwdin formula and Thouless theorem. We derive by an elementary method the infinitesimal or Löwdin projection operators and its integral representation to be useful for the projection of Slater determinant.

math-ph↗
arXiv2009

Angular momentum and the polar basis of harmonic oscillator

In this paper we follow the Schwinger approach for angular momentum but with the polar basis of harmonic oscillator as a starting point. We derive by a new method two analytic expressions of the elements of passage matrix from the double polar basis to 4- dimensions polar basis of the harmonic oscillator. These expressions are functions of the modules of magnetic moments. The connection between our results and the results derived by the group theory of Laguerre polynomials is found. We determine a new expression for these elements in terms of magnetic moments in the general case. We deduce from these expressions the symmetries of 3j symbols. A new generating function of the Clebsh-Gordan coefficients, functions of the modules of magnetic moments are found. We prove that the generating function of recoupling coefficients 3nj for the polar basis are the same in the Schwinger's approach therefore the polar basis of harmonic oscillator may be a starting point to study the angular momentum.

math-ph↗
arXiv2008

The two-dimensional hydrogen atom in The momentum representation

The analytic expression of the momentum representation in terms of the associated Legendre function is determined by a direct integration of Fourier transform of the wave function of coordinates using the Levi-Civita transformation and the generating function method. A new generating function for the associated Legendre functions are obtained.

math-ph↗
arXiv2008

On the hydrogen wave function in Momentum-space, Clifford algebra and the Generating function of Gegenbauer polynomial

Using the quadratic transformation and the generating function method we Perform the Fourier transformation of the wave function of coordinates of hydrogen atom and we find the analytic expression of the wave function in momentum space. We derive the matrix elements between the basis to 4-dimensions and integral representation of the generating functions of Gegenbauer polynomials. We find a relationship between a class of Clifford algebra and the generating functions of these polynomials.

math-ph↗
arXiv2008

On the Euler angles for the classical groups, Schwinger approach and Isoscalar factors for SU (3)

We establish recurrences formulas of the order of the classical groups that allow us to find a generalization of Euler's angles for classical groups and the invariant measures of these groups. We find the generating function for the SU(2) subset of SU(3) basis in the Fock-Bargmann space and a new basis of SU(3). This new basis is eigenfunction of the square of kinetic moment in product spaces of spherical harmonics. We generalize the generating function of SU (2) and we find invariant polynomials of SU (3) which are elements of the basis of SU (6). Using the above results we deduce the method of calculation isoscalar factors. We expose this method and we give the generating function for a particular case. Finally we determine the generating function of the elements of the representation matrix of SU(3) and we derive the analytical expression of these elements.

math-ph↗