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Heisenberg Uncertainty Principle for Momentum and Position Coordinates Referred to the Joule-Lenz Relation for Energy and Time

Stanisław Olszewski

Published Jul 25, 2017• 0 citations
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Abstract

The quantum aspects of the Joule-Lenz law for the dissipation of energy imply for small transition energies the validity of the formula △E △t = h where △t is the transition time of an electron particle. The present paper points out that a similar relation △px △x = h is valid for a particle enclosed in a one-dimensional potential box on condition the momentum change △px concerns two neighbouring quantum states. A corresponding change of the particle position coordinate △x can be not so small. The relation for △px and △x obtained for the one-dimensional case can be found to exist also in a three-dimensional system of the hydrogen atom taken as an example.

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The quantum aspects of the Joule-Lenz law for the dissipation of energy imply for small transition energies the validity of the formula △E △t = h where △t is the transition time of an electron particle. The present paper points out that a similar relation △px △x = h is valid for a particle enclosed in a one-dimensional potential box on condition the momentum change △px concerns two neighbouring quantum states. A corresponding change of the particle position coordinate △x can be not so small.
The relation for △px and △x obtained for the one-dimensional case can be found to exist also in a three-dimensional system of the hydrogen atom taken as an example.
The paper presents a non-probabilistic approach to the time interval associated with the energy emission produced by the electron transition in a quantum system. The calculations were performed for the hydrogen atom and the electron particle in a one-dimensional potential box. In both cases, the rule of conservation of the electron momentum has been applied. The results, limited to the time intervals of transitions between two neighbouring quantum energy levels, occur to be much similar to those obtained earlier with the aid of the Joule-Lenz energy emission theory.
The Joule-Lenz law for a classical expense of energy is transformed into a formula representing a quantummechanical invariant composed of the interval of energy connected with an electron transition and the corresponding interval of transition time between two quantum levels.Time and energy enter the invariant formula on an equal footing, moreover the time intervals converge with the time periods characteristic for the examined quantum systems.These properties imply to consider the time intervals as quanta of time having character similar to that possessed by the energy.Another result of the transformation of the Joule-Lenz law is the time rate of energy of the quantum transitions.This rate is calculated on a fully non-probabilistic way.When examined for the hydrogen atomic spectrum taken as an example, the obtained quantum rate is by many orders larger than a classical transition rate.
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.
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.
Definitions of the mechanical parameters entering the Bohr model of the hydrogen atom allowed us to calculate the time intervals connected with the electron transitions between the nearest-neighbouring energy levels in the atom. This is done in a strictly non-probabilistic way. The time results are compared with those derived earlier on the basis of the classical Joule-Lenz law for the energy emission adapted to the case of the electron transfer in the quantum systems. A similar formalism has been next applied to the harmonic oscillator and a particle moving in the one-dimensional potential box.
Quantum aspects of the Joule-Lenz law for the dissipation energy have been studied. In the first step, in an analysis of the energy-time principle of uncertainty, this gives a lower limit of the time interval and an upper limit of the energy interval which can be admitted in a quantum transition process. Moreover, for the low energy excitations, the transition time between the levels is found to be close to the oscillation time periods characteristic for these levels. A reference obtained among the transition time Δt, transition energy ΔE and the Planck constant h indicates that Δt should approach approximately the time period of the electromagnetic wave produced in course of the transition.