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Rigorous Derivation of the Uncertainty Relation and Research on Typical Instances

Xiong Zhi-hua

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

This paper proposes rigorous derivation of the uncertainty relation and analyzes this theory through some typical examples that particle momentum and particle location are measured at the same time.

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Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.
Quantum theory predicts an inherent joint unpredictability for some pairs of measurements. For example, Heisenberg showed that the more precisely the position of a quantum particle is known, the less precisely its momentum can be known and vice versa. Uncertainty relations (URs) are mathematical expressions quantifying the constraints between the output probability distributions of the given sets of measurements. Typically, URs are expressed in terms of uncertainty quantifiers such as entropies. Based on an information-theoretic approach, we discovered a characterization that unifies all uncertainty quantifiers and thus, generalizes a large class of URs into a single framework. We also prove new URs that are fundamentally different from typical URs in that they are fine-grained; i.e. they set restrictions directly on the output probability distributions without using any particular uncertainty quantifiers. We used Majorization theory and other techniques such as matrix analysis to prove our fine-grained uncertainty relations.
The axiomatic theory of quantum first-kind measurements is developed in a rigorous form based on five Postulates. The measurement theory for an observable with a continuous spectrum is given in a rigged Hilbert space. This approach also describes measurements with non-ideal initial conditions. It yields the survival effect in the position measurement of particles. It is also found that there is no such survival effect in the momentum measurement of particles. These Postulates of axiomatic theory yield the survival effect, which violates the Heisenberg uncertainty relation. This theoretical result is demonstrated by the wave function with a minimum of position and momentum uncertainty of the particle. The survival effect leads to essential corrections to the uncertainty relations. These modified uncertainty relations can also be used for the experimental verification of the survival measurement effect.