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Journal of Hebei Polytechnic University

A simply induced method of electronic track angular momentum formula

Zhan‐Xin Zhang

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

This paper takes hydrogen atom as an example, according to the differential equation. If the radial part R(r) of wave function Ψ of electron move has nothing to do with electronic revolves round core, through simple elementary calculation, we can obtain the electronic track angular momentum formula.

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In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the later cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrodinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
A discrete wave mechanical treatment of the hydrogen atom is extended to deal with states involving nonzero angular momentum. Only the radial portions of the wave vectors are covered. It is predicted that there is a nonzero minimum distance between the electron and the nucleus; this threshold distance increases with increasing angular momentum. Appropriate finite difference equations are formulated. The states with angular momentum exhibit the same degeneracy as do corresponding energy levels obtained from solutions of Schrödinger's equation.