Quantum mechanics phase-space structures

In this section we describe the dynamics and geometry associated with phase space structures governing reaction dynamics. While the emphasis in this review is on quantum mechanics, the phase space structure that we describe forms fhe classical mechanical "skeleton" on which the quantum mechanical theory is supported. Here we merely summarize the basic results. 

W is one quantum analogue of a classical phase space probability distribution function. One can, in fact, formulate quantum mechanics in terms of W instead of /. Now W(R,P,t=0) shown in Fig. 2b appears as a two dimensional Gaussian centred on R=6.75 a.u. and P = 0 a.u. It contains little information about the underlying classical phase space structure (Fig. 2a) except for the fact it is peaked where, classically, one has a significant area of quasibound motion. One sees no effects of the resonance islands, for example, because the phase space area occupied by these islands is less than Planck s constant h = 2tc a.u.. 

The study of the signatures of classical chaos in the quantum mechanical description of a general system is too complex for us to undertake at present. However, the phase space structure of a classical system that is exclusively defocussing is simpler than that of a general system. In particular, in an exclusively defocussing system the quasiperiodic motions of type (i) are absent. Examples of exclusively defocussing systems are the elastic collisions of a point particle with an assembly of hard discs or hard spheres or, indeed, any hard objects with smooth convex boundaries. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

The reason why the condition of Eq. (275) yields the breakdown of the correspondence between quantum and classical mechanics is evident. The quantum wave function can be identified with a trajectory if it is sharp enough, namely if U(t) < 2(f). When the width of the wave function becomes as large as the width of the channel, within which the wave function moves, the wave function motion starts to depend on the structure of the surrounding phase space, and the correspondence is broken. Using a geometrical model making E(t) decrease exponentially with time [122], the condition of Eq. (275) is shown to occur at time t = tB, where... 

According to quantum mechanics, isolated molecules do not have a finite boundary, but rather fade away into the regions of low electron density. It has been well established, however, from properties of condensed matter and molecular interactions, that individual molecules occupy a finite and measurable volume. This notion is at the core of the concept of molecular structure. 33 A number of physical methods yield estimations of molecular dimensions. These methods include measurements of molar volumes in condensed phases, critical parameters (lattice spacings and bond distances), and collision diameters in the gas phase. 34 From these results, one derives values of atomic radii from which a number of empirical molecular surfaces can be built. Note that the values of the atomic radii depend on the physical measurement chosen. 35-i37... 

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