  Investigation on foundational aspects of quantum statistical mechanics
recently entered a renaissance period due to novel intuitions from quantum
information theory and to increasing attention on the dynamical aspects of
single quantum systems. In the present contribution a simple but effective
theoretical framework is introduced to clarify the connections between a purely
mechanical description and the thermodynamic characterization of the
equilibrium state of an isolated quantum system. A salient feature of our
approach is the very transparent distinction between the statistical aspects
and the dynamical aspects in the description of isolated quantum systems. Like
in the classical statistical mechanics, the equilibrium distribution of any
property is identified on the basis of the time evolution of the considered
system. As a consequence equilibrium properties of quantum system appear to
depend on the details of the initial state due to the abundance of constants of
the motion in the Schr\"odinger dynamics. On the other hand the study of the
probability distributions of some functions, such as the entropy or the
equilibrium state of a subsystem, in statistical ensembles of pure states
reveals the crucial role of typicality as the bridge between macroscopic
thermodynamics and microscopic quantum dynamics. We shall consider two
particular ensembles: the random pure state ensemble and the fixed expectation
energy ensemble. The relation between the introduced ensembles, the properties
of a given isolated system, and the standard quantum statistical description
are discussed throughout the presentation. Finally we point out the conditions
which should be satisfied by an ensemble in order to get meaningful
thermodynamical characterization of an isolated quantum system.
