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Molecular Physics

S-matrix theory and the density matrix formalism

M. Roberts, W. E. Hagston

Published Aug 1, 1979• 3 citations
Publisher

Abstract

A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Møller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.

DOI
10.1080/00268977900101871

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Journal of Physics A Mathematical and General•Sep 11, 1984•2 citations
A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Møller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.
A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.
A recently developed method [Semkat et al., Phys. Rev. E 59, 1557 (1999); Kremp et al., in Progress in Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000), p. 34] for incorporating initial binary correlations into the Kadanoff–Baym equations (KBE) is used to derive a generalized T-matrix approximation for the self-energies. It is shown that the T-matrix obtains additional contributions arising from initial correlations. Using these results and taking the time-diagonal limit of the KBE, a generalized quantum kinetic equation in binary collision approximation is derived. This equation is a far-reaching generalization of Boltzmann-type kinetic equations: It self-consistently includes memory effects (retardation, off-shell T-matrices) as well as many-particle effects (damping, in-medium T-matrices) and spin-statistics effects (Pauli-blocking).
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
For pt.I see ibid. vol.16, p.3987-4000, (1983). The determinantal formalism for solving the time-dependent Schrodinger equation of the density matrix has been developed, in I, assuming a constant external perturbation. The important case of an harmonic perturbation is presented here. Detailed expressions are derived for diagonal and nondiagonal elements of the response, and the trace conservation theorem is shown to be satisfied at all times and any order. The results are again applied to the simple illustrative quantum system considered in I.