Potential well analogy

We have obtained a broad range of detailed quantitative results relating to universality in the absence of the electron-phonon interaction by various theoretical techniques. The methods include perturbation theory, the coherent potential approximation (CPA), field theory, path integral methods, numerical calculations and the potential well analogy. The results include the density of states, the nature of the wave functions, the mean free path, the energy dependent... 

Recent advances based on the coherent potential approximation and the potential well analogy suggest that, in spite of the great complexity of the potential felt by an electron moving in an amorphous semiconductor, the electronic structure of the latter possesses a certain universality. As a result (and to a first approximation) only a few parameters matter. The situation is analoguous to that of a crystalline semiconductor, where a single quantity - the effective mass - allows one to bypass the complexity of the crystalline potential. 

In the present illustration, the polymer PMF is written as a series of merged harmonic potentials in analogy to the double-well potential of Straub et al. ... 

Polymer identification starts with a series of preliminary tests. In contrast to low molecular weight organic compounds, which are frequently satisfactorily identified simply by their melting or boiling point, molecular weight and elementary composition, precise identification of polymers is difficult by the presence of copolymers, the statistical character of the composition, macromolecular properties and, by potential polymeric-analogous reactions. Exact classification of polymers is not usually possible from a few preliminary tests. Further physical data must be measured and specific reactions must be carried out in order to make a reliable classification. The efficiency of physical methods such as IR spectroscopy and NMR spectroscopy as well as pyrolysis gas chromatography makes them particularly important. 

On the other hand, if the rate of formation of a doublet is much smaller than its rate of dissociation, the doublet is unstable. It will be shown later that the unstable doublets reach a dynamic equilibrium with the singlets in an extremely short time (of the order of the time scale of dissociation). The equilibrium concentration of these unstable doublets is small and depends upon the relative magnitudes of the rates of formation and dissociation. Since the dissociation rate of a doublet decreases rather dramatically with increasing particle size (because of the rapid increase in the depth of the interaction potential well with increasing particle size), there exists a critical particle size above which the coagulated particle pair is stable, i.e., its rate of formation is much greater than its rate of dissociation. This critical particle size is analogous to the critical cluster size... 

We have tried to represent this concept pictorially in Fig. 20, where we transformed the complex pattern of reaction paths illustrated in Scheme 11 into a network of potential wells and barriers, by substituting each intermediate species with a potential well of different depth, separated from adjacent intermediate species by specific activation barriers of different heights. In other words. Fig. 20 is the evolution of the analogous Fig. 3 obtained by increasing the number of degrees of freedom of the reaction from 1 to 2, and where the termination processes have not been taken into consideration. 

As noted earlier, the stored electrons in each potential well are converted to a digital value by an analog-to-digital (A/D) converter, with the gain, )/, usually stated as e /count or electrons per analog to digital unit (e /ADU) ... 

Wave-particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, then we can apply the resnlts of classical electromagnetic theory to particles. By analogy, the probability is the sqnare of the amplitnde. Zero-point energy is a con-seqnence of the Heisenberg nncertainty relation all particles bound in potential wells have finite energy even at the absolnte zero of temperature. 

---