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Journal of Physics A Mathematical and General

Entropic uncertainty relations for the infinite well

Vladimír Majerník, Lukáš Richterek

Published Feb 21, 1997• 53 citations
Publisher

Abstract

We calculate the entropic and standard uncertainty relations for the position and momentum of the infinite potential well as functions of its quantum states. We show that the entropic uncertainty relations express more adequately the uncertainty principle than the standard ones which use the dispersions as measures of the uncertainties for two non-commuting observables.

DOI
10.1088/0305-4470/30/4/002

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We calculate the entropic and standard uncertainty relations for the position and momentum of the infinite potential well as functions of its quantum states. We show that the entropic uncertainty relations express more adequately the uncertainty principle than the standard ones which use the dispersions as measures of the uncertainties for two non-commuting observables.
This paper deals with the standard and entropic uncertainty relations for the finite-potential well with different length and potential depth. We calculate the position and momentum dispersions and entropies that are dependent on the quantum states of the finite well and determine the corresponding standard and entropic uncertainty relations. Then we calculate the entropy of occurrence for the mass point inside and outside the well. We discuss the results and determine a class of probability distributions with a parameter for which the standard deviation becomes infinite, independent of the value of this parameter.
The uncertainty principle in quantum physics for non-commuting observables can be quantitatively expressed in various mathematical forms. Well known are common Heisenberg-like uncertainty relations in which the product of dispersions of two non-commuting observables is used. An alternative form of uncertainty relations represents the so-called entropic uncertainty relations in which the sum of entropies of the considered non-commuting observables is employed. The aim of this paper is to show that the entropic uncertainty relations exhibit several interesting mathematical properties and can be taken as an adequate alternative formulation of the uncertainty principle in quantum mechanics. To compare the mathematical properties of the common and entropic uncertainty relations we present some simple quantum systems with two non-commuting observables. Dedicated to Dr B Mamojka on the occasion of his fiftieth birthday. Abstrakt. Princip neurcitosti pro nekomutující veliciny v kvantové fyzice mze být matematicky vyjádren r formou. Dobre známé jsou relace neurcitosti Heisenbergova typu, v nichz se vyuzí vá soucinu disperzí dvou nekomutujících fyzikálních velicin. Alternativou jsou tzv. entropické relace neurcitosti, v nichz vystupuje soucet entropií uvazovaných velicin. Cílem tohoto clánku je ukázat, ze tyto relace mají nekolik z matematického hlediska zajímavých vlastností a mohou být pouzity k alternativnímu vyjádrení principu neurcitosti v kvantové mechanice. Srovnání vlastností Heisengergových a entropických relací neurcitosti je demonstrováno na jednoduchém kvantovém systému s dvema nekomutujícími velicinami.
We present the entropic uncertainty relations for multiple measurement settings which demonstrate the uncertainty principle of quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory, and can be proven by a unified method. Our results recover some well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. The bounds of those relations which quantify the extent of the uncertainty take concise forms and are easy to calculate. Those uncertainty relations might play important roles in the foundations of quantum theory. Potential experimental demonstration of those entropic uncertainty relations is discussed.