Quantum mechanics bifurcation

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

MSN. 151.1. Prigogine, Why irreversibility The formulation of classical and quantum mechanics for non-integrable systems, Int. J. Bifurcation and Chaos 5, 3-16 (1995). 

A quantum mechanical proof of the bifurcation of the decay modes is... 

Founargiotakis, M., Farantos, S.C., Contopoulos, G., and Polymilis, C. (1989). Periodic orbits, bifurcations, and quantum mechanical eigenfunctions and spectra, J. Chem. Phys. 91, 1389-1401. 

The situation, one hundred years on, could hardly be more different. The interpretation of quantum mechanics, which came to replace the Newtonian system, is as hotly disputed as ever and the common ground with the theory of relativity remains elusive and vague. The reason for the discord must lie somewhere in the transition from the classical to the new non-classical paradigm. What is proposed here, is to retrace the steps that led to the emergence of the new theoretical models, in an attempt to identify the point of conceptual bifurcation. 

It has been shown recently that the vibrational spectra of HCP [33-36], HOCl [36-39], and HOBr [40,41] obtained from quantum mechanical calculations on global ab initio surfaces can be reproduced accurately in the low to intermediate energy regime (75% of the isomerization threshold for HCP, 95% of the dissociation threshold for HOCl and HOBr) with an integrable Fermi resonance Hamiltonian. Based on the analysis of this Hamiltonian, this section proposes an interpretation of the most salient feature of the dynamics of these molecules, namely the first saddle-node bifurcation, which takes place in the intermediate energy regime. 

Saddle-node bifurcations taking place for the reasons just described have been observed for HOBr [41], HOCl [36,38,39], and HCP [34-36]. For HOBr and HOCl, the stable PO bom at the saddle-node bifurcations is called [D] for dissociation, because this PO stretches along the dissociation pathway and scars OBr- or OCl-stretch quantum mechanical wavefunctions (see Fig. lie of Ref. 38, Figs. 3b and 3g of Ref. 41, or Section III.B). In the case of HCP, the stable PO born at the bifurcation is better called [I], for isomerization, because this PO stretches along the isomerization pathway and scars bending quantum mechanical wavefunctions (see Figs. 6b and 6d of Ref. 35 or Figs. 7b and 7d of Ref. 36). 

The quantum mechanical probability density is compared to an ensemble of classical trajectories in Fig. 32. Both quantities carry the same dynamical features, which, again, supports the intuitive picture evolving from LCT theory. It is seen that a bifurcation occurs, where two wavepackets move out of phase with each other (and likewise do sets of classical trajectories). This then has the consequence that, upon reaching the continuum, the rotational motion is not directional to 100%. Rather, for the present parameters, one finds a ratio of 2.2 in favor of the counterclockwise rotation. 

To understand the internal molecular motions, we have placed great store in classical mechanics to obtain a picture of the d5Uiamics of the molecule and to predict associated patterns that can be observed in quantum spectra. Of course, the classical picture is at best an imprecise image, because the molecular dynamics are intrinsically quantum mechanical. Nonetheless, the classical metaphor must surely possess a large kernel of truth. The classical structure brought out by the bifurcation analysis has accounted for real patterns seen in wavefunctions and also for patterns observed in spectra, such as the existence of local mode doublets, and the... 

There are also approaches [M, M and M] to control that have had marked success and which do not rely on quantum mechanical coherence. These approaches typically rely explicitly on a knowledge of the internal molecular dynamics, both in the design of the experiment and in the achievement of control. So far, these approaches have exploited only implicitly the very simplest t5q)es of bifurcation phenomena, such as the transition from local to normal stretch modes. If further success is achieved along these lines in larger molecules, it seems likely that deliberate knowledge and exploitation of more complicated bifurcation phenomena will be a matter of necessity. 

In Section IV.D, the crisis is studied in classical mechanics. In this section, we study how it manifests itself in quantum mechanics. There are two phenomena where the outcomes of the crisis can be observed. The first is bifurcation in reaction paths, which directly corresponds to the changes of the connections. The second is multiexponential decay in the predissociation processes, which reflects the existence of multiple dissociation paths with different decay times. We will further discuss the possibility of observing phenomena related to the crisis. 

The quantum mechanical treatment of a three-dimensional atom-diatom reactive system is one of the main subjects of theoretical chemistry [1]. About a decade ago when the first numerical results for the H + H2 reactions appeared in print [2] it seemed that the problem was solved. However, difficulties associated with numerical instabilities and with the bifurcation into two nonsyrametric product channels slowed progress with this kind of treatment. This situation caused a change in the order of priorities whereas previously most of the effort was directed toward developing algorithms for yielding "exact cross sections, now it is mostly aimed at developing reliable approximations. 

Figure 4 shows two typical trajectories which trace the quantum mechanical wave-packet motion. They are started on the symmetric stretch line with different initial momenta pointing into the exit channels. The two trajectories represent dissociation into H + OD and D + OH, respectively. What was found for the quantum dynamics is more clearly demonstrated by the classical trjectories the H + OD dissociation is faster and the oscillations of both trajectories around the minimum energy path clearly shows the vibrational excitation of the fragments. Indeed, if we compute the time-evolution of the bondlength expectation values th d) from the bifurcated packets it is found that they resemble closely the classical trajectories (as can be expected from Ehrenfest s theorem). The wave-packet motion shows that the dissociation proceeds as can be anticipated classically. 

The splitting of an SL of the quantum mechanical cmrent densities J and at a branching point in can be smdied within the framework of bifurcation theory of dynamical systems, see [94—96] for an introduction to the subject. Accordingly, let us consider a system of first-order differential equations in matrix form... 

P. B. Middleton and R. E. Wyatt, Quantum mechanical study of a reaction path bifurcation model, Chem. Phys. Lett. 21 57 (1973). 

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