Unbound motion

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

In order to make connections between chaotic properties and statistical behavior in reaction dynamics, one must first define chaotic properties of open systems, since all chemical reactions involve unbounded motion in at least one coordinate. A way of linking chaotic behavior in bound systems to that in open systems was discussed previously for classical unimolecular decay. However, in the quantum case, we do not attempt a similar link but rather establish the circumstances under which chaos in closed systems implies statistical behavior in open systems. 

There are also some infinitely rare motions with parabolic-elliptic original and/or final evolution, with an escape in the past and a capture in the future - or inversely, or even with an oscillating motion of the first type i.e with an unbounded motion composed of an infinite number of larger and larger loops in the ooCDoo annulus, but with all pericenters in the second annulus ABCD. 

Figure 7.19a is a pictorial description of Ev) for an outer wall curve crossing. In phase space, bound motion in the uth vibrational level appears as an ellipse in the harmonic approximation. Motion on a linear unbound potential is represented as a parabola. The shaded area is 2 increases with u for E > Ec Whenever the value of maximum value [except for the first maximum, v = 0, at which Eq. (7.6.11) is invalid] (Child, 1980b). 

Separatrix The line or surface in the phase space of a particle that distinguishes between two classes of trajectories as, for example, between boimded and unbounded motion. 

Translation of a quantum mechanical particle in a container corresponds to a bound state with the energy quantized. Translation of a particle, such as an atom or molecule, in an unrestricted space (no walls and no potentials) is an unbound motion. The mechanics of such a free particle have certain similar elements in both the quantum and classical pictures. In both, the energy may vary continuously. In chemistry then, energy storage depends on the nature of the mechanical system and on whether or not there are bound states. We will revisit all of the quantum mechanical systems discussed earlier later in the book. 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

It is more probable that the phenomenon consists of a tunneling of electrons into the metal. We shall first consider the case of an unbound neutral atom or molecule near a surface, choosing H for simplicity. Since electronic motion is much faster than that of nuclei, we may consider the... 

Every one knows that heat can produce motion. That it possesses vast motive-power no one can doubt, in these days when the steam-engine is everywhere so well known. .. The question has often been raised whether the motive power of heat is unbounded, whether the possible improvements in steam-engines have an assignable limit—a limit which the nature of things will not allow to be passed by any means whatever or whether, on the contrary, these improvements may be carried on indefinitely. .. We propose now to submit these questions to a deliberate examination. 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

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