Trapped trajectory

E. Poliak, P. Pechukas, Transition states, trapped trajectories, and classical bound states embedded in the continuum, J. Chem. Phys. 69 (1978) 1218. 

In the time-dependent picture, resonances show up as repeated recurrences of the evolving wavepacket. Resonances and recurrences reveal, in different ways, the same dynamical effect, namely the temporary excitation of internal motion within the complex. In the context of classical mechanics, the existence of quantum mechanical resonances is synonymous with trapped trajectories performing complicated Lissajou-type motion before they finally dissociate. The larger the lifetime, the more frequently the wavepacket recurs to its starting position, and the narrower are the resonances. 

Because the periodic orbits are highly unstable in the present example (i.e., the slightest distortion destroys the periodicity and leads to dissociation) the trapped trajectories recur only once after returning to their origin they start again, with a new set of initial conditions, and most probably will dissociate in the second round. 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

Fig. 1.3. Irregular motion in box C. This box is derived from box R by adding a totally reflecting disk at the centre of R. (a) A complicated but exiting trajectory is produced for the laimch angle p = 0.69. (b) Dynamically trapped trajectory for a laimch angle close to p 0.692.

The application of symbolic dynamics to the three-disk scattering system helped us to focus right away on the essential questions (i) Is it possible to have an infinite number of scatterings (ii) Are there trapped trajectories, and, if yes, how many ... 

Since the set A of trapped orbits is not reachable from the outside, it repells scattering trajectories that eventually exit the system. Exiting scattering trajectories, which at some point in their time evolution may be close to a trapped trajectory, will eventually be repelled from the trapped trajectory and scatter off to infinity. This is the reason why the set A of unreachable trapped trajectories is also called a repeller. Repeller... 

Fig. 1. Illustration of scattering by three discs of radius a, whose centres are separated by the distance R. The solid line is the periodic trapped trajectory labelled 123. The dashed line is a scattering trajectory which escapes from the interaction region after six reflections by a disc...

The three-disc system is an example of a hyperbolic scattering system. For a scattering system to be hyperbolic it must contain an uncountable set of trapped trajectories which are all unstable, i.e. the Liapunov exponent describing the rate of exponential growth of infinitesimal deviations from a given trajectory is positive for each trapped trajectory. Hyperbolicity also requires that the trapped trajectories are uniformly unstable, meaning that there is a finite positive lower bound to all Liapunov exponents [39]. These conditions are fulfilled for the three-disc system [29]. 

There is a Cantor set of trapped trajectories which show up in the deflection functions or in a phase space portrait of the scattering time delays or survival times. This is indicated in Fig. 8 showing the phase space structure at a held strength = 2, where there are no islands of stability. The initial conditions of those trajectories with exactly two zeros are marked black, the white regions in between correspond to trajectories with three or more zeros. A break-down of these regions according to the number of zeros reveals self-similar fractal structure [62]. 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

D. Effects of the Democratic Centrifugal Force Inducing Mass-Balance Asymmetry and Trapping Trajectories Around the Transition State... 

It is seen that the bifurcations are nearly removed and that now the motion is much more in favor of a counterclockwise rotation. In fact, one finds that 83% of the rotational wavepackets move in this direction. The classical trajectories exhibit a dynamics similar to that of the quantum densities. This is true for the ones moving freely but also for the trapped trajectories which do not have enough energy to escape the potential wells for a discussion of similar features and their relation to femtosecond pump-probe signals, see Ref. 214. 

Figure Bl.7.15. Stability diagram for ions near the centre of a quadrupole ion trap mass spectrometer. The enclosed area reflects values for and that result in stable trapping trajectories (reproduced with permission of Professor R March, Trent University, Peterborough, ON, Canada).

P. Pechukas, E. Poliak, Trapped trajectories at the boundary of reactivity bands in molecular collisions, J. Chem. Phys. 67 (12) (1977) 5976-5977. 

E. Poliak and P. Pechukas,. Chem. Phys., 69, 1218 (1978). Transition States, Trapped Trajectories, and Classical Bound States in the Continuum. 

In the more dynamical theory of Pechukas and Poliak the dividing surfaces at the flux extrema of the present statistical model become periodic classical trajectories that oscillate across the potential valley, i.e., trapped trajectories". It is interesting to see that many aspects of this more detailed dynamical treatment appear, albeit approximately, in a purely statistical model. 

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