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arXiv:0903.0708

Angular momentum and the polar basis of harmonic oscillator

Mehdi Hage-Hassan

Published Mar 4, 2009• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.0903.0708

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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.
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.
Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d mm' j ( theta ) and which are written as t mm jj' ( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
The cylindrically and the spherically symmetric harmonic oscillators are considered by means of matrix techniques in a basis that is appropriate for each symmetry. For example, the spherically symmetric oscillator is described by the radial and angular momentum quantum numbers. The energy eigenvalues and the matrix elements of the promotion-demotion operators are found in each case. By expressing the operators in curvilinear coordinates, the promotion-demotion operators are used to deduce the form of the corresponding wave functions.
Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.