Schrodinger-Langevin equation

The Schrodinger-Langevin equation and the quantum master equation... 

In section 2 it was mentioned that a damping term in the Schrodinger equation violates the conservation of probability. It is possible, however, to compensate this loss by adding artificially a suitable fluctuating term. The result will be an equation for if/ resembling the classical Langevin equation. 

Our Schrodinger-Langevin equation (5.2) involves no T or equivalent parameter it is therefore not possible to obtain a Schrodinger equation with damping. 

Exercise. The Schrodinger-Langevin equation (5.3) implies that the average of any Heisenberg operator (1.24) obeys... 

Thus we have found the general form of the quantum master equation that corresponds to the Schrodinger-Langevin equation (5.3). 

We have arrived at the quantum master equation (5.6) heuristically via the Schrodinger-Langevin equation (5.3). It turns out, however, that (5.6) has a firmer foundation it can be proved mathematically on the basis of the following three general conditions concerning the evolution of p. 

One solution to this problem is that implemented in the dynamical equations (6) and (7), which couple the classical Langevin equation for the atoms in the protein-water system to the stationary Schrodinger equation for the quantum excitations (thus ensuring that only quantum eigenstates are considered). One limitation of this solution is that it is only valid when the quantum excitation responds very fast to any changes in the classical conformation, something which is assumed to be true here. 

Quantum mechanics and statistical mechanisms did not change. Starting from the basic equations - Schrodinger, Langevin, Fokker-Planck - one has developped very good methods of solution as the simulation methods which take into account most of the observed features and, perhaps, some imaginary ones. 

Participation of flie surface phonons in the energy transfer process with translational degrees of freedom can be accounted for by two theoretical approaches based on the theory of the generalized Langevin equation and on quantum or semi-classical solutions of flie Schrodinger equation. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The product implies integration over q as in (1.1) and also summation over the components of ip if there are more than one. Evidently the average norm is conserved when 2U = V V. Thus we have found a Langevin extension of the Schrodinger equation ... 

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