Center for Theoretical Physcis of the Polish Academy of Sciences, Warsaw

Quantum uncertainty relations for infinite-dimensional Hilbert space are recalled.
Apart of the standard, 1927 approach of Heisenberg-Kennard,
based on the product of variances of the probability distribution in two different representations, we advocate an alternative 1975
approach of Bialynicki-Birula and Mycielski, in which sum of two entropies characterizing both distributions is investigated.
In the case of a finite-dimensional Hilbert space
we present the 1983 bounds of Deutsch and 1988 of Maassen and Uffink
and discuss possibilities to improve them. In particular,
a generic case of two orthogonal measurements performed in two bases
related by a random unitary matrix is analyzed.
Complementary part of the talk is focused on properties of quantum wave
functions - solutions of the Schroedinger equation for model systems.
If a quantum system is described in an infinite Hilbert space,
such wave functions can display genuinely fractal properties,
which in physical systems can be observed only up to a certain accuracy.
However, such nowhere differentiable functions are by no mean exceptional,
as they form a prevalent set in the space of all continuous functions.