Stochastic dynamical systems Schrodinger equation

The pure state of the molecular environment is never precisely known. Since the dynamics of the molecular pure state depends on this unknown environment s (pure) state, one gets a stochastic dynamics for the molecular pure states. For a two-level system-with its pure states describable by a Bloch sphere—the situation is illustrated in Fig. 8. The dynamics of some given molecular initial state is governed not only by the Schrodinger equation, but also by external influence. Depending on the (pure) state of the environment, we reach different final molecular states. Usually only probabilistic predictions can be given (and no information about the precise trajectory of pure states, i.e., the trajectory on the Bloch sphere). 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, zi and z, which satisfy the Markovian equations of motion... 

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Schrodinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underl)dng dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. 

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