Kicked molecule

To anticipate the result of pulsed excitation of a superposition state, note from Eqs. (6.66) and (6.68) that the Hamiltonian is strictly periodic in time. We denote the time evolution operator associated with one period T as F. Although it is not possible to give an explicit form of F in the kicked molecule case, the existence j of this formal solution yields a stroboscopic description of the dynamics,... 

The plan of Chapter 5 is the following. In order to get a feeUng for the dynamics of the kicked molecule, we approximate it by a one-dimensional schematic model by restricting its motion to rotation in the x, z) plane and ignoring motion of the centre of mass. In this approximation the kicked molecule becomes the kicked rotor, probably the most widely studied model in quantum chaology. This model was introduced by Casati et al. in 1979. The classical mechanics of the kicked rotor is discussed in Section 5.1. Section 5.2 presents Chirikov s overlap criterion, which can be applied generally to estimate analytically the critical control parameter necessary for the onset of chaos. We use it here to estimate the onset of chaos in the kicked rotor model. The quantum mechanics of the kicked rotor is discussed in Section 5.3. In Section 5.4 we show that the results obtained for the quantum kicked rotor model are of immediate... 

At first glance, the kicked molecule experiment sketched in Fig. 5.1 does not appear to be a system worthy of much attention. The set-up is simple, there are no comphcated boundary conditions, and noise effects are neglected. But its simpHcity notwithstanding, it turns out that the classical as well as the quantum dynamics of the system sketched in Fig. 5.1 are very complicated, and cannot in either case be solved analytically in the presence of a strong driving field. 

In order to extract the essence of the dynamics of the kicked molecule, we consider a simple model constructed by replacing the molecule with a two-dimensional dipole (see Fig. 5.2). Moreover, we replace the sequence of finite-width pulses provided by the pulse generator (see Fig. 5.1) by a train of zero-width -function kicks. Thus, the dipole is perturbed by a... 

File 5-3 Output From a Stochastic Search of the n-Pentane Molecule in MM3. The input structure was subjected to 50 kicks. ... 

Colloidal particles experience kicks from the surrounding atoms or molecules of the solvent. This leads to Brownian dynamics in colloidal suspensions (Fig. 14). The study of dynamics is challenging as, of course, first the equilibrium of the system has to be understood. One often knows the short-time dynamics that govern the system and is interested in long-time properties. 

You have replied that your molecule has a 5-fold axis of rotation. Verify that it also has 15 binary axes and ten ternary axes. Note that it belongs to one of the icosahedral groups. If you play soccer, consider the ball. Before you kick it, look at it. What is its symmetry ... 

Desperate, you try methanol, one of the most polar solvents. It is really held strongly to the adsorbant. So it comes along and kicks the living daylights out of just about all the molecules in the mixture. After all, the methyl alcohol is more polar, so it can move right along and displace the other molecules. And it does. So, when you evaporate the methanol and look, all the mixture has moved with the methanol, so you get one spot that moved, right with the solvent front. 

Since the ester is relatively insensitive to water (and probably other similar impurities [la, 2b, c]) but the propagating ions are excessively so [lb, 28] we think that in the earlier work the ions formed by the slow dissociation of the styrene-solvated ester were inactivated as quickly as they were formed ions became evident as chain-carriers only when the depletion of monomer removed the stabilising coordinated monomer molecules from the ester and this then ionised very rapidly, producing the final very fast reaction-kick observed by Gandini and Plesch [1]. 

To include the effects of collisions on the rotational motion part of any of the above C(t) functions, one must introduce a model for how such collisions change the dipole-related vectors that enter into C(t). The most elementary model used to address collisions applies to gaseous samples which are assumed to undergo unhindered rotational motion until struck by another molecule at which time a randomizing "kick" is applied to the dipole vector and after which the molecule returns to its unhindered rotational movement. 

Referring again to the metal spheres of submicroscopic dimensions, one point becomes clear. The smaller they are ( microns), the more they react to the thermal kicks from the ions and water molecules of the electrolyte they take off on a random walk through the solution. Large ( centimeters) spheres also exchange momentum with the particles of the solution, but their masses are huge compared with those of ions or molecules, so that the velocities resulting (to the spheres) from such collisions are essentially zero. 

This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. 

In activated complex theory, two molecules are pictured as approaching each other and distorting as they meet. In the gas phase, that meeting and distortion is the collision of collision theory. In solution, the approach is a zigzag walk among solvent molecules, and the distortion might not take place until after the two reactant molecules have met and received a particularly vigorous kick from the solvent molecules around them (Fig. 13.19). In either case, the collision or the kick does not imme-... 

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