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AND THE GENERALIZED UNCERTAINTY RELATIONS

Ken K uriyama

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Sesquilinear forms induced by a positive linear form on a * -algebra are introduced. Some relations among them are investigated and then the inequality related to the uncertainty principle is shown in general situation. As a by-product, GNS construction theorem by the right regular representation is constructed by the * -antihomomorphism.

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•Jan 1, 2014•0 citations
Sesquilinear forms induced by a positive linear form on a * -algebra are introduced. Some relations among them are investigated and then the inequality related to the uncertainty principle is shown in general situation. As a by-product, GNS construction theorem by the right regular representation is constructed by the * -antihomomorphism.
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.
We analyze extremal measurement error relations for time and frequency parameters of optical pulses, which are compared with the uncertainty relation. It is shown that the product of the mean-square measurement error limit for a single photon corresponds to the minimum value of ˙/2 in the quantum uncertainty relation for time-energy (coordinate-momentum) and decreases with deviation from a Gaussian pulse shape, for which the information entropy is maximal. The error relation is in the form of the uncertainty relation with the inequality reversed.