We have to verify that p goes over into the density function in the classical limit and that it has ergodic properties. The goal is to turn this quantity p into the classical density function in phase space. Let us denote the square of this amplitude by p( n 1, n 2., n N). Particular wavefunction of the basic set participates in the actual wavefunction of the system. The natural variable which we have is the amplitude with which a Let us suppose we have a set of wave functions which depend parametrically on a set of quantum numbers n 1, n 2., n N. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a conceptually similar, but mathematically different way. Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. In the classical framework we compute the partition function of the system in order to evaluate all possible thermodynamic quantities. The density matrix formalism was developed to extend the tools of classical statistical mechanics to the quantum domain. On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector. The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau. He provided in this work a theory of measurement, where the usual notion of wave collapse is described as an irreversible process (the so called von Neumann or projective measurement). John von Neumann rigorously established the correct mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik.

Von neumann entropy