The kicked molecule

In this section we investigate the physics of impulsively perturbed diatomic molecules. These provide a real physical system whose characteristics are expected to be close to the ones established for the kicked rotor. We establish that the dynamical effects exhibited by the quantum kicked rotor can be demonstrated experimentally with diatomic molecules within the possibilities of present day technology. 

Following Townes and Schawlow (1975), we characterize the orientation of a diatomic molecule by two angles, 9 and (p. Then, the stationary 

The conditions of normalizability and single-valuedness require that the spectrum of (5.4.4) is discrete. It is given explicitly by 

In a realistic experiment we cannot work with zero-width kicks . Any realistic electric field pulse has a finite width. Therefore, we replace the periodic S function drive in (5.3.2) by an array of finite-width pulses according to 

We will now integrate the time dependent Schrodinger equation for the wave function (t) of a Csl molecule. We expand the wave function according to 

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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... 

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. 

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-... 

Although the reductionist argument is of obvious validity, the inverse process of constructionism is impossible. This philosophy assumes that the properties of more complex systems can be predicted from those of a simpler one. By this logic theoretical chemists of the 20th century have persistently tried to reconstruct chemical behaviour from the fundamental equations of wave mechanics. To date the most powerful computers on the planet have failed consistently to reconstruct even the most fundamental property in all of chemistry, namely the structure of a molecule. Computations, known as quantum chemistry, all have to rely on the kick-start of an assumed molecular structure. 

Fig. 5.1. The kicked Csl molecule schematic sketch of a proposed experimental set-up. The electric field driving the molecule is generated by a voltage V t) provided by a pulse generator.

Our first impression of Fig. 5.4(a) is one of regularity and periodicity. The phase-space structures seem to be periodic in I with a period of 2it. This is indeed so and is easily proved by inspection of the mapping (5.1.6). Since (5.1.66) refers to an angle variable, it is to be understood as an equation modulo 27t. Changing I by multiples of 2i clearly leaves (5.1.66) invariant. The same transformation also leaves (5.1.6a) invariant, since the 2i increments in I cancel on both sides. Thus, the rotor phase space is 2i periodic in 1. The periodicity in I is often used in the hterature to reduce the study of the kicked rotor to the unit cell 0,1) [0, 27t] X [0, 27t]. We do not use this technique here since the phase space appropriate for our particular atomic physics problem, the kicked Csl molecule, is the phase-space cyhnder with unrestricted values of I, and not the restricted phase space consisting only of a single unit cell. 

The shape of the pulse (5.4.28) is shown in Fig. 5.11. The shape (5.4.28) was chosen because it may be possible to actually synthesize pulses resembling (5.4.28) with an array of phase-locked microwave generators (see discussion below). Kicking the Csl molecules with the pulses (5.4.28) results in the excitation of rotational states. 

The physics behind this relation is the fluctuation-dissipation theorem the same random kicks of the surrounding molecules cause both Brownian diffusion and the viscous dissipation leading to the frictional force. It is -instructive to calculate the time scale t required for the particle to move a... 

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