Derivation of the quantum Langevin equation

The phenomenological approach to the Langevin equation breaks down for quantum systems. It is necessary to take into account explicitly an external bath, which causes the damping and fluctuations. The question then is whether there exists an autonomous equation of motion for the system itself, with or without a memory kernel. Some simple cases are listed and for one case the derivation of Langevin's equation is carried out. It is compared to the exact result, which leads to the following conclusions. The equation is true only to the lowest order in the interaction between system and bath. In general there is a memory kernel but in two special cases the damping reduces to a simple proportionality with a friction coefficient (''Ohmic case''). (C) 1997 Elsevier Science B.V.