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arXiv:1604.00991

Theta-Quantization of the Orbital Angular Momentum Operator by means of Noncommutative Geometry

Takeo Miura

Published Mar 20, 2016• 0 citations
arXivHTMLPDF

Abstract

The eigenfunctions and eigenvalues of orbital angular momentum operator on noncommutative lattice for a circle poset by theta-quantization are constructed, and it is demonstrated that they are equivalent to those of the conventional quantum mechanics.
DOI
10.48550/arxiv.1604.00991

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The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
Eigenfunctions and eigenvalues of the operator of the square of the angular momentum are studied. It is shown that neither from the requirement for the eigenfunctions be normalizable nor from the commutation relations it is possible to prove that the eigenvalues spectrum is a set of only integer numbers (in units ℏ=1\hbar=1ℏ=1). We present regular, normalizable eigenfunctions with the non-integer eigenvalues thus demonstrating that a non-integer angular momentum is admissible from the theoretical viewpoint.
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