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American Journal of Physics

Density Matrix Representations

John D. Patterson

Published Dec 1, 1962• 6 citations
Publisher

Abstract

The relations between the ordinary density matrix, the density matrix in the coordinate representation, the Löwdin density matrices, the Dirac density matrix, and other density matrices are traced. The roles of pure and mixed cases, equilibrium and nonequilibrium situations, and single- and many-body representations are considered.
DOI
10.1119/1.1941845

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This chapter delves into the concept of density matrices, a fundamental tool in quantum mechanics for describing the state of a many-body system. Density matrices provide a compact and powerful representation of quantum states, encompassing both pure and mixed states. The study of one-body and two-body density matrices is particularly important for understanding the behavior of many-body systems, especially in contexts like quantum gases, solid-state physics, and quantum chemistry. These matrices encapsulate information about particle distributions and correlations, enabling the derivation of key properties such as energy, momentum, and entropy.
A realistic interpretation of Schroedinger and Dirac equations for density matrices is proposed, in which the difference between the position arguments of the density matrix is considered as an objective extra space dimension. "Particle" solutions are found, which are perfectly localized both in position space and in momentum space (the position and momentum operators commute in the density matrix representation); definitions for all observable quantities are given and the values associated to the "particle" solutions are the correct ones, both for the non-relativistic and the relativistic case. Finally, a non-linear interaction (the electromagnetic one) is introduced in an attempt to single out the "particle" solutions of the Dirac equation from all other solutions; the dynamical evolution of the electromagnetic field is described by the classical (unquantized) Maxwell equations.
A realistic interpretation of Schroedinger and Dirac equations for density matrices is proposed, in which the difference between the position arguments of the density matrix is considered as an "objective" extra space dimension. "Particle" solutions are found, which are perfectly localized both in position space and in momentum space (the position and momentum operators commute in the density matrix representation); definitions for all observable quantities are given and the values associated to the "particle" solutions are the correct ones, both for the non-relativistic and the relativistic case. Finally, a non-linear interaction (the electromagnetic one) is introduced in an attempt to single out the "particle" solutions of the Dirac equation from all other solutions; the dynamical evolution of the electromagnetic field is described by the classical (unquantized) Maxwell equations.