EXACT GENERALIZED FOKKER-PLANCK EQUATION FOR ARBITRARY DISSIPATIVE AND NONDISSIPATIVE QUANTUM-SYSTEMS

We start from a density matrix equation in its most general form. It comprises the action of external fields on the system, internal interactions, as well as the action of dissipative mechanisms (heat-baths or reservoirs), which may be Markoffian or non-Markoffian. We then define a distribution function of a type introduced previously by Haken, Risken, Weidlich for atoms. This distribution function, f, which is now formulated quite generally with aid of projection operators, Pik, establishes a connection between the Pik's and classical variables vik. By means of f it is possible to exactly calculate all quantum mechanical expectation values by pure c-number procedures. If the basic density matrix equation is Markoffian, it is even possible to calculate all time-ordered multitime averages by c-number procedures using f, as had been demonstrated by Haken, Risken and Weidlich. In the present paper we derive in an explicit way an exact c-number partial differential equation for f. It contains derivatives of arbitrarily high order. In important classes of problems, it can be reduced to an ordinary FokkerPlanck equation, however. Our new equation has many applications, e.g. in the quantum theory of lasers, nonlinear quantum optics, spinresonance, and spin-wave-theory, as will be demonstrated in forthcoming papers. We wish to thank Prof. W. Weidlich and Dipl. Phys. H. Vollmer for several valuable discussions. In addition, H. Vollmer has kindly checked our calculations. © 1969 Springer-Verlag.