Quantum-classical correspondence

A comprehensive discussion of wavepackets, classical-quantum correspondence, optical spectroscopy, coherent control and reactive scattering from a unified, time dependent perspective. 

In conclusion it appears that the basic facts about the classical/quan-tum correspondence of classically chaotic atoms and related atomic systems is by now well understood, at least on the conceptual phenomenological level. For the near future we see the main research direction in strengthening the classical-quantum correspondence using the ever more powerful semiclassical methods that are currently being developed and will soon be a part of the tool-kit of modern atomic physics. 

Kim, J.-H. and Ezra, G.S. (1991). Periodic orbits and the classical-quantum correspondence for doubly-excited states of two-electron atoms, in Proceedings of the Adriatico Conference on Quantum Chaos, eds. H. Cerdeira et al. (World Scientific, Hong Kong). 

The clearest results have been obtained for classical relaxation in bound systems where the full machinery of classical ergodic theory may be utilized. These concepts have been carried over empirically to molecular scattering and decay, where the phase space is not compact and hence the ergodic theory is not directly applicable. This classical approach is the subject of Section II. Less complete information is available on the classical-quantum correspondence, which underlies step 4. This is discussed in Section III where we introduce the Liouville approach to correspondence, which, we believe, provides a unified basis for future studies in this area. Finally, the quantum picture is beginning to emerge, and Section IV summarizes a number of recent approaches relevant for a quantum-mechanical understanding of relaxation phenomena and statistical behavior in bound systems and scattering. 

Longstanding interest in classical-quantum correspondence results from the fact that many processes involving the dynamics of nuclei occur in a regime suitable to classical and semiclassical approximations. Many recent studies have focused on the embedding of classical dynamics in appropriate quantumlike formulas to produce semiclassical approximations.49 Although useful for semiclassical developments, these approaches have several disadvantages for studies of the classical-quantum correspondence. The most obvious of these is the absence of the analogue to the probabilty amplitude ip in classical... 

In this section we advocate a far more advantageous route to studying conceptual features of the classical-quantum correspondence, and indeed for each mechanics independently, in which phase space distributions are used in both classical and quantum mechanics, that is, classical Liouville dynamics50 in the former and the Wigner-Weyl representation in the latter. This approach provides, as will be demonstrated, powerful conceptual insights into the relationship between classical and quantum mechanics. The essential point of this section is easily stated using similar mathematics in both quantum and classical mechanics results in a similar qualitative picture of the dynamics. 

Note that these auxiliary conditions, particularly (31b), are the reason why the nontraditional (e.g., complex) nature of the underlying classical eigen-distributions is suppressed in classical mechanics. Such features are, however, vital for an understanding of the classical-quantum correspondence from the distributions viewpoint. 

Our emphasis has been on the fundamental role of the eigendistributions of Lc and La, and it is within this eigenfunction basis that classical-quantum correspondence assumes its most simple form. At present the correspondence is clearly understood for stationary and nonstationary eigendistributions in regular systems and for stationary eigendistributions in the chaotic case. Further work is necessary to clarify the picture for nonstationary chaotic eigendistributions. 

It is instructive to examine properties of spatial correlation functions for specific cases. The stadium-billiard problem, discussed in previous sections in the context of classical-quantum correspondence, is a useful example. There is, in this case, no potential within the boundary, that is, the boundary alone induces ergodicity. Therefore, the q integration is readily done, yielding,... 

The classical-quantum correspondence also results in useful scaling laws for chaotic states. For the stadium problem the classical correlation functions scale as (2m ) l/z, a feature which is solely a consequence of classical ergodicity. As shown in Fig. 17, quantum states labeled chaotic (according to the aperiodic nature of their correlation function) do obey this scaling relation. Specifically, Fig. 17 displays the correlation lengths (A,/2), defined as... 

One particular approach to classical-quantum correspondence deals with the behavior of coherent states as h approaches zero. It is interesting to note that this approach then deals with states which are, for all h, physically acceptable in both quantum and classical mechanics. 

Clearly, a Wigner swarm of classical trajectories is a valuable exploratory tool for estimating quantum-mechanical branching ratios. The computer times involved are more than 100 times shorter than for the quantum-mechanical calculations. For larger masses and more degrees of freedom the disparity between classical and quantum calculation times will become even more pronounced. At the same time, the classical-quantum correspondence should be even better than illustrated here, for larger mass systems. 

The recent development of internal coordinate quantum Monte Carlo has made it possible to directly compare classical and quantum calculations for many body systems. Classical molecular dynamics simulations of many body systems may sometimes overestimate vibrational motion due to the leakage of zero point energy. The problem appears to become less severe for more highly connected bond networks and more highly constrained systems. This suggests that current designs of some nanomachine components may be more workable than MD simulations suggest. Further study of classical-quantum correspondence in many body systems is necessary to resolve these concerns. 

Of course, many of the essential features of a liquid are preserved in classical simulations. What determines these liquid properties more than the total energy is energy differences with respect to thermal motion, external forces, and so on. In the context of classical-quantum correspondence, it is important to note that any liquid is an unbound system and so the zero point energy problem has far less significance in considerations of liquid structure than in the problems... 

One important remark is in order. That is, although quantum phenomena have been observed in molecular systems, we possess only the very qualitative traditional rules regarding conditions under which quantum effects predominate. Specifically, if the initial state involves large classical actions and the initial state is one that is allowed classically, then quantum effects tend to be small. Considerably more work is necessary, however, before more quantitative, predictive statements can be made and before our understanding of classical/quantum correspondence in bound molecular systems is complete. 

An elegant method to introduce the quantum particles motion s equation, called - at non-relativistic level - the Schrodinger equation, consists in employing the classical-quantum correspondences in Table 3.1. Thus, for the conservation of the total energy of a particle under the action of an external potential V(x) there is equivalently obtained that the classical form of energy conservation (Putz, 2006)... 

To understand the classical/quantum correspondence for systems with two or more degrees of freedom. 

Section II below outlines possible approaches for understanding the classical dynamics of vdW systems with three degrees of freedom. Section III will address aspects of the classical/quantum correspondence. Section IV consists of some brief concluding remarks. 

Phase Space Dynamics and Energy Levels, Classical/ Quantum Correspondence, and RRKM Theory... 

One of the advantages of classical mechanics over quantum mechanics is that it is possible to know everything about a system. However, since the real world is quantum mechanical, it is necessary to relate classical and quantum properties properly. Although most applications of classical trajectories are well within the safe zone of the classical approximation, it is still necessary to interpret the classical-quantum correspondence. The calculation of state-resolved cross sections or rate coefficients provides an illustration of the problem since it requires that the continuous classical results be partitioned into quantized levels. ... 

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