Semiclassical approaches advantages

In general, in the study of systems at equihbrium the main advantage of the semiclassical approaches is that they can be utilized in simulation work by following essentially the same techniques as in the classical case (i.e., one works with Ng particles). Moreover, whenever they can be applied, semiclassical approaches can work excellently and represent a substantial saving in computational effort as compared to the exact PI xP-calculations. On the negative side, even the best semiclassical approaches cannot cope with very large quantum diffraction effects. Actually, they do not capture all the information needed to characterize the quantum system (e.g., some g (r) structures cannot be computed). In the literature of this field there are different names for these pair potentials (e.g., classical effective potentials, semiclassical effective potentials, quantum effective pair potentials, etc.). The absence of external fields will be assumed in this section, and the system will be ruled by the Hamiltonian... 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

Summary. An efficient semiclassical optimal control theory for controlling wave-packet dynamics on a single adiabatic potential energy surface applicable to systems with many degrees of freedom is discussed in detail. The approach combines the advantages of various formulations of the optimal control theory quantum and classical on the one hand and global and local on the other. The efficiency and reliability of the method are demonstrated, using systems with two and four dimensions as examples. 

As this approach deals with a set of classical trajectories, its numerical cost remains reasonable for multidimensional systems. Contrary to the classical approach, which controls only the averaged classical quantities, the present semiclassical method can control the quantum motion itself. This makes it possible to reproduce almost all quantum effects at a computational cost that does not grow too rapidly as the dimensionality of the system increases. The new approach therefore combines the advantages of the quantum and classical formulations of the optimal control theory. 

Thus we have formally, and exactly, converted the master equation to a Schroedinger equation. This has the substantial advantage that we can apply well-known approximations in quantum mechanics to obtain solutions to the master equation. In particular we refer to the W.K.B. approximation valid for semiclassical cases, those for which Planck s constant formally approaches zero. The equivalent limit for (3.8) is that of large volumes (large munbers of particles). Hence we seek a stationary solution of (3.8), that is the time derivative of Px X,t) is set to zero, of the form... 

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