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

TL;DR

This paper introduces a generating function method with diverse applications in quantum, nuclear, and classical group theory, providing solutions to complex problems and deriving key invariants and representations.

Contribution

It develops a versatile generating function approach that simplifies solving problems in physics and group theory, including invariants, symbols, and representations.

Findings

Generated the harmonic oscillator's generating function

Derived invariants of SU(2) and 3-j, 6-j symbols

Expressed the Hamiltonian in terms of quasi-bosons

Abstract

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 Schr\"odinger equation from Hamilton-Jacobi formalism. We present these applications in independent chapters.