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Journal of Yancheng Institute of Technology

Simple Deduction and Correct Understanding of Uncertainty Relation

Weijun Yu

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

According to de Broglie wave hypothesis,based on one dimensional infinite wall and by using standing wave condition,distribution of momentum of a particle in one dimensional infinite wall is deduced.With reference to the definition of uncertainty,position uncertainty and momentum uncertainty are calculated and uncertainty relation of one dimensional infinite wall is deduced.Correct understanding uncertainty relation is discussed.

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•Mar 2, 2021•0 citations
According to de Broglie wave hypothesis,based on one dimensional infinite wall and by using standing wave condition,distribution of momentum of a particle in one dimensional infinite wall is deduced.With reference to the definition of uncertainty,position uncertainty and momentum uncertainty are calculated and uncertainty relation of one dimensional infinite wall is deduced.Correct understanding uncertainty relation is discussed.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Abstract In 1925, de Broglie hypothesized that any material particle has an associated wave with /h p . Electron diffraction seems to support that Hypothesis. But then, the electron at rest will have infinite wavelength, and infinite wave phase velocity. This says that for a material particle, the de Broglie relation does not hold. Failed attempts to save the postulate, kept the flawed relation, and modified the waves into train waves, pilot waves, probability waves, … to name a few. We keep the de Broglie waves unchanged, and modify the relation. First, we observe that the Planck energy E h used by de Broglie defines virtual electromagnetic waves. Consequently, for any particle, the virtual electromagnetic wavelength is / /c h mc , and / /h m c . Refinement of de Broglie argument, indicates that /h mmay be x , Heisenberg’s uncertainty in the particle location. de Broglie’s later analysis supports this interpretation, and we offer an explanation to particle diffraction as a consequence of Heisenberg’s uncertainty. We apply / /h m c to obtain the dispersion relations for the de Broglie virtual waves.
In this paper,the uncertainty in positon,the momentum probability amplitude and momentum probability density's recurrence formulas,the uncertainty in momentum and the uncertain relation of the equality form momentum-position of one-dimensional harmonic oscillator were deduced.This paper also pointed out that the lower limit of one-dimensional harmonic oscillator's ΔpΔx is exactly the lower limit of /2 which given by the general form of the uncertain relation,however the lower limit of one-dimensional infinite well's ΔpΔx is a little bigger than /2,so the lower limit of a quantum system's ΔpΔx is needn't equal to /2.
It is possible to prove the principle of certainty in a different way from the theory of relativity, de Broglie particle wave duality, the principle of uncertainty. The principle of certainty is the formula for measuring the position and momentum of a particle at the same time. If the theory of relativity, the principle of uncertainty is correct, then the principle of certainty must be considered correct. Because it is possible to prove the principle of certainty from the principle of uncertainty, it is also possible to prove the principle of certainty from the theory of special relativity. Again the idea of the principle of certainty comes from the de Broglie particle wave duality principle. Max Planck's radiation formula can be proved from the principle of certainty.