Introduction

Statistical Mechanics is a theory that establishes the connection between the observed properties of systems with many degrees of freedom and the microscopic quantum mechanical properties of the elementary constituents of the systems (e.g., electrons, atoms and molecules). Such a theory builds upon the description of matter provided by quantum mechanics and provides the molecular foundation of Thermodynamics. Contrary to evaluating the precise N-particle dynamics of macroscopic systems, Statistical Mechanics describes the properties of systems in terms of the statistics of possible microscopic states. The description of measurements is, therefore, given in terms of the ensemble average of expectation values associated with the quantum states that constitute such an ensemble.

Macroscopic systems consist of an enormously large number of degrees of freedom (e.g., $\sim 10^{23}$ electrons, atoms or molecules), so many degrees of freedom that in practice it is impossible to prepare such systems in a well defined microscopic quantum state (i.e., in a pure quantum state). Instead, they are usually prepared in thermodynamic states (i.e., in a statistical mixtures of quantum states) characterized by a few physical quantities (e.g., the temperature, the pressure, the volume and the number of particles).

In order to describe macroscopic systems in terms of quantum mechanics it is, therefore, necessary to incorporate into the formalism the incomplete information about the state of the system. The most natural approach is provided by appealing to the concept of probability. This can be accomplished by introducing the density operator, a very useful mathematical tool which facilitates the simultaneous application of the postulates of quantum mechanics and the results of probability calculations.