The role of the probability distribution density is taken by the density matrix in a quantum system. The density matrix satisfies the quantum Liouville theorem. Also, similarly we can define different ensembles described by different density matrices. Using a semi-classical single-particle approach examples of several systems are considered such as an ideal gas, the harmonic oscillator and the specific case of a two-level system is considered and negative temperatures are discussed as population inversion. In the case of identical particles the quantum nature of the problem is emphasized, and two classes of systems, free bosons and free fermions, are discussed in detail, including a photon gas. The phenomenon of Bose-Einstein condensation, that results from a “statistical” interaction, is treated. The black-body radiation and the birth of quantum mechanics are referred. Also, the significance of the Fermi surface in a fermionic system is emphasized.