Double well potential system

Isomerization Rate Constants for Symmetric Double-Well Potential Systems Defined in Table XIV... 

From these observations, we learn that even simple double-well potential systems demonstrate a variety of dynamics, depending on the shape of potential functions. Note that this sort of question—that is, what information in the potential function is necessary and sufficient conditions which make the system chaotic—dates back to just the beginning of the study of chaos in few-degrees-of-freedom systems. 

B.I. Modify quantum.lD.m to compute the lowest-energy states for the double-well potential system shown in Figure 3.8, with the parameters... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

We hope to have convinced the reader by now that the tunneling centers in glasses are complicated objects that would have to be described using an enormously big Hilbert space, currently beyond our computational capacity. This multilevel character can be anticipated coming from the low-temperature perspective in Lubchenko and Wolynes [4]. Indeed, if a defect has at least two alternative states between which it can tunnel, this system is at least as complex as a double-well potential—clearly a multilevel system, reducing to a TLS at the lowest temperatures. Deviations from a simple two-level behavior have been seen directly in single-molecule experiments [105]. In order to predict the energies at which this multilevel behavior would be exhibited, we first estimate the domain wall mass. Obviously, the total mass of all the atoms in the droplet... 

In order to interpret the observation of reactions which have low efficiencies, we have suggested a double well potential surface model (8) illustrated in Figure 2. This is the simplest model which is consistent with available data. At the low pressures typically used in ICR, long collision times ensure that the system contains its initial total energy throughout the reaction. The efficiency for an exothermic reaction is given by /(k j + k ). Passage over the central barrier (k )... 

Ferroelectricity is a highly collective phenomenon involving the long-range internal electric field, and occurs only when the size of the system is large enough. Yet local physics does play a major role, as is evidenced by the relevance of the lattice anharmonicity that develops a local double-well potential. 

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. 

A different way, developed extensively by Schwartz and his coworkeis, - is to use approximate quantum propagators, based on expansions of the exponential operators. These approximations have been tested for a number of systems, including comparison with the numerically exact results of Ref 38 for the rate in a double well potential, with satisfying results. 

The rate of tautomerization can be related to a simple model describing the transformation of the BPS to IS by considering the motion of the proton in a double-well potential, with the system either in the N—H or the NH+ configuration of the two moieties. In this model, the rate is given by the tunneling expression ... 

The two-dimensional PES shown in Figure 8.17 (as well as in Figures 8.3b and 8.7c) is typical of internal rotation coupled to inversion of the other part of the system. This situation is also realized in methylamine inversion, where the rotation barrier is modulated not by a harmonic oscillation but by motion in a double-well potential. The PES for these coupled motions can be modeled as follows ... 

To show how the junction rule works, consider the above example of tunneling in the double-well potential. In this case we have two nodes connected by just one tunneling path. Let the starting position of the system be in the left well with the ground-state wave function P1 = C il> (r) (Q) are assumed to be normalized, and C is the amplitude in the left well, so that I C I2 is the probability to find the system in this well. The corresponding tail of the WKB ground-state wave function under the barrier should decrease with Q exponentially,... 

The Duffing Equation 14.4 seems to be a model in order to describe the nonlinear behavior of the resonant system. A better agreement between experimentally recorded and calculated phase portraits can be obtained by consideration of nonlinear effects of higher order in the dielectric properties and of nonlinear losses (e.g. [6], [7]). In order to construct the effective thermodynamic potential near the structural phase transition the phase portraits were recorded at different temperatures above and below the phase transition. The coefficients in the Duffing Equation 14.4 were derived by the fitted computer simulation. Figure 14.6 shows the effective thermodynamic potential of a TGS-crystal with the transition from a one minimum potential to a double-well potential. So the tools of the nonlinear dynamics provide a new approach to the study of structural phase transitions. 

The existence of two stable steady states for the same values of the externally controlled constraints is known as bistability . A mechanical analogy to such a situation is the double well potential shown in Fig. 2. (No potential function is known to exist for open chemical systems that would play the role of the gravitational potential energy in... 

Consider two enantiomorphous molecules, R (right-handed) and L (left-handed), isolated from their surroundings and from external fields but not from each other. That is, they are allowed to interact. In the quantum-mechanical treatment of this system, as two particles in a one-dimensional symmetric double-well potential, the two states are degenerate in energy and are related to wavefunctions TR and 4Y localized in the two potential wells. Superposition of these wavefunctions yields ground and first excited states F+ and P ... 

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