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Journal of Modern Physics

De Broglie’s Velocity of Transition between Quantum Levels and the Quantum of the Magnetic Spin Moment Obtained from the Uncertainty Principle for Energy and Time

Stanisław Olszewski

Published Jan 1, 2014• 7 citations
Publisher

Abstract

The De Broglie's approach to the quantum theory, when combined with the conservation rule of momentum, allows one to calculate the velocity of the electron transition from a quantum state n to its neighbouring state as a function of n.The paper shows, for the case of the harmonic oscillator taken as an example, that the De Broglie's dependence of the transition velocity on n is equal to the n-dependence of that velocity calculated with the aid of the uncertainty principle for the energy and time.In the next step the minimal distance parameter provided by the uncertainty principle is applied in calculating the magnetic moment of the electron which effectuates its orbital motion in the magnetic field.This application gives readily the electron spin magnetic moment as well as the quantum of the magnetic flux known in superconductors as its result.

DOI
10.4236/jmp.2014.518198

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The De Broglie's approach to the quantum theory, when combined with the conservation rule of momentum, allows one to calculate the velocity of the electron transition from a quantum state n to its neighbouring state as a function of n.The paper shows, for the case of the harmonic oscillator taken as an example, that the De Broglie's dependence of the transition velocity on n is equal to the n-dependence of that velocity calculated with the aid of the uncertainty principle for the energy and time.In the next step the minimal distance parameter provided by the uncertainty principle is applied in calculating the magnetic moment of the electron which effectuates its orbital motion in the magnetic field.This application gives readily the electron spin magnetic moment as well as the quantum of the magnetic flux known in superconductors as its result.
In the first step, the Ehrenfest reasoning concerning the adiabatic invariance of the angular orbital momentum is applied to the electron motion in the hydrogen atom. It is demonstrated that the time of the energy emission from the quantum level n+1 to level n can be deduced from the orbital angular momentum examined in the hydrogen atom. This time is found precisely equal to the time interval dictated by the Joule-Lenz law governing the electron transition between the levels n+1 and n. In the next step, the mechanical parameters entering the quantum systems are applied in calculating the time intervals characteristic for the electron transitions. This concerns the neighbouring energy levels in the hydrogen atom as well as the Landau levels in the electron gas submitted to the action of a constant magnetic field.
A transformation of the electron states—say those enclosed in a potential box—into the de Broglie waves done in the paper, enabled us to calculate the energy change between two quantum levels as a function of the specific heat and difference of the temperature between the states. In consequence, the energy difference and that of entropy between the levels could be examined in terms of the appropriate classical parameters. In the next step, the time interval necessary for the electron transition between the levels could be associated with the classical electrodynamical parameters like the electric resistance and capacitance connected with the temporary formation of the electric cell in course of the transition. The parameters characterizing the mechanical inertia of the electron were next used as a check of the electrodynamical formulae referring to transition.
The mechanical angular momentum and magnetic moment of the electron and proton spin have been calculated semiclassically with the aid of the uncertainty principle for energy and time.The spin effects of both kinds of the elementary particles can be expressed in terms of similar formulae.The quantization of the spin motion has been done on the basis of the old quantum theory.It gives a quantum number n = 1/2 as the index of the spin state acceptable for both the electron and proton particle.In effect of the spin existence the electron motion in the hydrogen atom can be represented as a drift motion accomplished in a combined electric and magnetic field.More than 18,000 spin oscillations accompany one drift circulation performed along the lowest orbit of the Bohr atom.The semiclassical theory developed in the paper has been applied to calculate the doublet separation of the experimentally well-examined D line entering the spectrum of the sodium atom.This separation is found to be much similar to that obtained according to the relativistic old quantum theory.
The time of the energy emission between two neighbouring electron levels in the hydrogen atom has been calculated first on the basis of the quantum aspects of the Joule-Lenz law, next this time is approached with the aid of the electrodynamical parameters characteristic for the electron motion in the atom. Both methods indicate a similar result, namely that the time of emission is close to the time period of the electromagnetic wave produced in course of the emission. As a by-product of calculations, the formula representing the radius of the electron microparticle is obtained from a simple combination of the expressions for the Bohr magnetic moment and a quantum of the magnetic flux.
In the first step, the Joule-Lenz dissipation energy specified for the electron transitions between two neighbouring quantum levels in the hydrogen atom has been compared with the electromagnetic energy of emission from a single level. Both the electric and magnetic vectors entering the Pointing vector of the electromagnetic field are referred to the one-electron motion performed along an orbit in the atom. In the next step, a similar comparison of emission rates is performed for the harmonic oscillator. Formally a full agreement of the Joule-Lenz and electromagnetic expressions for the energy emission rates has been attained. The case of the electron oscillator allowed us to perform a more direct comparison of the quantum approach to the emission rate with the classical electromagnetic theory.