Double well theory

Needless to say, tunneling is one of the most famous quantum mechanical effects. Theory of multidimensional tunneling, however, has not yet been completed. As is well known, in chemical dynamics there are the following three kinds of problems (1) energy splitting due to tunneling in symmetric double-well potential, (2) predissociation of metastable state through... 

In this chapter, we discuss double layer theory and how it can be incorporated into a geochemical model. We will consider hydrous ferric oxide (FeOOH //IFO), which is one of the most important sorbing minerals at low temperature under oxidizing conditions. Sorption by hydrous ferric oxide has been widely studied and Dzombak and Morel (1990) have compiled an internally consistent database of its complexation reactions. The model we develop, however, is general and can be applied equally well to surface complexation with other metal oxides for which a reaction database is available. 

It is thought that the micelle surface is undulating to some degree, so it is not a perfect entity. However, it is described well by double-layer theory, which requires some order. It is not easy to describe a micelle as nobody has ever seen one it is something that is projected from all its properties. 

The expressions for the depopulation factor as given in Eqs. 29 and 30 for the single and double well potential cases respectively, remain unchanged. This version of the turnover theory for space and time dependent friction has been tested successfully against numerical simulation data, in Refs. 68,137. 

This semiclassical turnover theory differs significantly from the semiclassical turnover theory suggested by Mel nikov, who considered the motion along the system coordinate, and quantized the original bath modes and did not consider the bath of stable normal modes. In addition, Mel nikov considered only Ohmic friction. The turnover theory was tested by Topaler and Makri, who compared it to exact quantum mechanical computations for a double well potential. Remarkably, the results of the semiclassical turnover theory were in quantitative agreement with the quantum mechanical results. 

From that time on, many causal theories were developed. In this work, we shall only refer to the causal theory proposed by de Broglie [2] and known as the double-solution theory. This theory, as well as Bohm s theory, are the most developed of all causal theories they are both able to explain and predict, practically all quantum phenomena. 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

Figure 6. Population decay of the initial state in a barrierless double-well system calculated using multilevel Redfield theory [25]. The vibrational frequency is 60 cm1. (----------)...

Hund, one of the pioneers in quantum mechanics, had a fundamental question of relation between the molecular chirality and optical activity [78]. He proposed that all chiral molecules in a double well potential are energetically inequivalent due to a mixed parity state between symmetric and antisymmetric forms. If the quantum tunnelling barrier is sufficiently small, such chiral molecules oscillate between one enantiomer and the other enantiomer with time through spatial inversion and exist in a superposed structure, as exemplified in Figs. 19 and 24. Hund s theory may be responsible for dynamic helicity, dynamic racemization, and epimerization. 

It is well-known that the traditional double layer theory is valid in a limited range of concentrations for monovalent electrolytes, but is much less valid for higher valency electrolytes.15 The traditional theory starts from the Poisson equation... 

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