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Journal of Anqing Teachers College

Some Notes on Uncertainty Relation

Gui Chuan-you

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

This article points out the imprecision of the eduction of uncertairnty relation from wave packet while the motion of microcosmic particles is determined by quantum statistical theorem.And it expounds the two sides of the meaning of uncertainty relation based on the explanation of two typical quantum phenomena.

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The mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Fejér average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Fejér average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the above-mentioned points.

Journal of Mathematical Physics•Jan 1, 2002•6 citations
The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.
Quantum uncertainty Spin I want to start this section with a description of particle spin. In addition to the usual properties of mass, charge, position and velocity, particles have a further parameter which describes their state of rotation. Physicists call this ‘spin’. We have already seen that in quantum physics it is impossible to measure simultaneously, in a precise way, both speed and position (recall the Heisenberg uncertainty relations). The same considerations apply to spin, which cannot be absolutely determined with arbitrary precision. The things that we can in principle find out are the rate of rotation (that is, the number of revolutions per second) and the component of this rotation in a given direction in space (for example, the angle of the axis of rotation relative to the direction of motion). It is important to understand that these two knowable quantities can take only certain quantised values. This is an aspect of quantum physics that we have already encountered when describing wave motion. We can use as an analogy the stable vibrations that can be sustained on a piano string: the basic note is generated when the centre of the string vibrates to its maximum extent while the two ends are (necessarily) fixed.
The principle of uncertainty is one of the essential principles of quantum mechanics as it outlines that there is a reflective limit to the measurement of microscopic particle motion. However it is impossible to measure at once and accurately both the position and momentum data of any microscopic particle no matter what means are used to do so. This paper starts by discussing the notion and mathematical derivation of the uncertainty principle as put forth by Heisenberg explaining the physical relevance of the principle. It introduces the natural uncertainty the principle introduces to the dynamics of particles in the microscopic world, and the significant part it plays in the theory of quantum mechanics. The article overviews the explanations of the principle on phenomena including atomic stability and scanning tunneling microscopes, and the interference of measurement actions on the states of particles and the shortcomings of measurement precision. According to the wave-particle duality of quantum mechanics, this article gives a required theoretical basis of microscopic phenomena. Likewise, the use of the research theory on the uncertainty principle by distinguished scholars in the physics fraternity helps the readers in achieving the knowledge in related subjects more easily.
The mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Fejér average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Fejér average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the above-mentioned points.