\nthe quantum calculation of the d.c. electrical conductivity for an electron system is reconsidered by using an improved form of the long-time solution of the density matrix. The starting time-evolution equation which is written as a linear system in the Liouville space of discrete quantum states, is first solved in terms of Cramer's determinants and next suitably adapted to tranport problems. The calculation of the statistical average of the many-body current density, in the presence of scattering potential, is carried out in the thermodynamic limit, using a characteristic property of collision matrices introduced earlier by van Hove as the so-called
delta-singularity. In contrast to previous methods based on kinetic equations solved by perturbations, Kubo's theory, or Mori type memory functions, the resolution of an integral equation, initially included in Cramer's formulae, is no longer required. Among its major advantages, the method succeeds in overcoming the well known zero frequency divergent behaviour of the conductivity. It also yields an explicit expression of an effective relaxation time which is just constructed out of those diagonal transition sequences responsible for self-energy effects in van Hove's theory. This expression which only assumes the randomization of phases of initial states can be used to get tractable expansions in powers of the collision potential. Its range of validity is limited by the effect of collisions on equilibrium statistics, which is shown to be negligible within the Peierls criterion,
hbar/taullmu,
tau being the relaxation time and
mu the Fermi energy. Finally, the comparison of the present approach with the elementary transport theory reveals that the relaxation time is close to the standard expression, even though the latter would be exactly recovered, in lowest order of collisions, only on disregarding divergent terms arising beyond second order.\n
We investigate the occurrence of exponential relaxation in a certain class of closed, finite systems on the basis of a time-convolutionless projection operator expansion for a specific class of initial states with vanishing inhomogeneity. It turns out that exponential behavior is to be expected only if the leading order predicts the standard separation of time scales and if, furthermore, all higher orders remain negligible for the full relaxation time. The latter, however, is shown to depend not only on the perturbation (interaction) strength, but also crucially on the structure of the perturbation matrix. It is shown that perturbations yielding exponential relaxation have to fulfill certain criteria, one of which relates to the so-called "Van Hove structure." All our results are verified by the numerical integration of the full time-dependent Schrödinger equation.