Steady-state quantum mechanics of thermally relaxing systems

A theoretical description of quantum mechanical steady states is developed. Applications for simple quantum mechanical systems described in terms of coupled level structures yield a formulation equivalent to time independent scattering theory. Applications to steady states of thermally relaxing systems leads to time independent scattering theory in Liouville space that is equivalent to the tetradic Green's function formalism. It provides however a direct route to derive particular forms of the Liouville equation applicable in steady-state situations. The theory is applied to study the conduction properties in the super-exchange model of a metal-molecule-metal contact weakly coupled to the thermal environment. The energy resolved temperature dependent transmission probability, as well as its coherent (tunneling) and incoherent (activated) parts, are calculated using the Redfield approximation. These components depend differently on the energy gap (or barrier), on the temperature and on the bridge length. The coherent component is most important at low temperatures, large energy gaps and small chain lengths. The incoherent component dominates in the opposite limits. The integrated transmission provides a generalization of the Landauer conduction formula for small junctions in the presence of thermal relaxation.