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.

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Life is dependent on the income of energy with low entropy and the disposal of energy with high entropy. On Earth, the low-entropy energy is provided by solar radiation and the high-entropy energy is disposed as infrared radiation emitted into the cold space. Here we turn the situation around and assume cosmic background radiation as the low-entropy source of energy for a planet orbiting a black hole into which the high-entropy energy is disposed. We estimate the power that can be produced by thermodynamic processes on such a planet, with a particular interest in planets orbiting a fast rotating Kerr black hole as in the science fiction movie {\em Interstellar}. We also briefly discuss a reverse Dyson sphere absorbing cosmic background radiation from the outside and dumping waste energy to a black hole inside.Comment: 8 pages, 6 figures. Replaced with the version published in Am. J. Phy

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Two black holes can merge to create a bigger black hole, thus increasing the entropy of the universe. Alternatively, they can be used as two heat reservoirs from which work can be extracted. We discuss a process during which two black holes are transformed into one while the total entropy is kept as constant. The resulting black hole has a smaller mass than the total mass of the input black holes and the mass difference is converted into work. Although the process will probably not be used within the next 1011 yr for energy production, we can speculate that it might be an energy source for those who might inhabit our universe after that. We discuss the basic thermodynamics of the proposed system.