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Journal of Kashgar Teachers College

Matrix Representation about Orbit Angular Momentum Operator and Eigenvector of Delectron in Hydrogen Atom

Liu Na

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

We have got the Matrix representation of delectron in hydrogen atom about orbit angular momentum operator and eigenvector in Lz representation,resolved the problem of eigenvector about the operators of Lx and Ly.

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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.
A new mathematical method is established to represent the operator, wave functions and square matrix in the same representation. We can obtain the specific square matrices corresponding to the angular momentum and Runge-Lenz vector operators with invoking assistance from the operator relations and the orthonormal wave functions. Furthermore, the first-order differential equations will be given to deduce the specific wave functions without using the solution of the second order Schrodinger equation. As a result, we will unify the descriptions of the matrix mechanics and the wave mechanics on hydrogen atom. By using matrix transformations, we will also deduce the specific matrix representations of the operators in the SO (4,2) group.
Elementary treatments of a single angular momentum having components Jk (k =1, 2, 3) usually go as far as discussions of the spectra of these operators and the derivations of their standard, i.e., J2, J3-diagonal, representatives. On the other hand, the standard representatives of their eigenvectors are not usually obtained, leaving J3 aside, for which the problem is trivial. In fact, this part of the theory is most often dealt with only after a lengthy excursion into the theory of matrix representations. The representatives in question are obtained, in the first part of this paper, by means of essentially elementary arguments. Once these representatives are known, the standard representative of the unitary operator U which generates the state vector of a system after it has undergone a rotation R from the state vector prior to the rotation are deduced; this representative comprises just the familiar matrix representations of the rotation group. Finally, the relevance of harmonic oscillator theory to the present problem is briefly considered.