Correspondence classical

As we shall see, in molecules as well as atoms, the interplay between the quantum description of the internal motions and the corresponding classical analogue is a constant theme. However, when referring to the internal motions of molecules, we will be speaking, loosely, of the motion of the atoms in the molecule, rather than of the fiindamental constituents, the electrons and nuclei. This is an extremely fundamental point to which we now turn. 

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

Many other cations besides the norbomyl cation have nonclassical structures. Scheme 5.5 shows some examples which have been characterized by structural studies or by evidence derived from solvolysis reactions. To assist in interpretation of the nonclassical stmctures, the bond representing the bridging electron pair is darkened in a corresponding classical stmcture. Not surprisingly, the borderline between classical stmctures and nonclassical stmctures is blurred. There are two fundamental factors... 

Pagano s exact solution for the stresses and displacements is too complex to present here. The corresponding classical lamination theory result stems from the equilibrium equations, Equations (5.6) to (5.8), which simplify to... 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

However, in all the rest of their approach, Robertson and Yarwood consider the slow mode Q as a scalar obeying simply classical mechanics, because they neglect the noncommutativity of Q with its conjugate momentum P. As a consequence, the logic of their approach is to consider the fluctuation of the slow mode as obeying classical statistical mechanics and not quantum statistical mechanics. Thus we write, in place of Eq. (138), the corresponding classical formula ... 

The results of the quantum simulations for cases A, B and C are shown in the lower two panels in Fig. 3. The corresponding classical phase portraits shown reinforce our inferences from the stability diagram no stabilization for A while larger islands exist for C as compared with B. However, the ionized fraction as calculated from the quantum evolution supports the contrary result that there is more stabilization for A as compared with B. Case C is the most stable which is at least consistent with the classical prediction. What is the origin of this discrepancy ... 

Quantum dynamics on graphs became an issue also in the context of quantum information. Aharonov et.al (1993) pointed out that a random quantum walk on one dimensional chains can be faster than the corresponding classical random walk. Since then, a whole field has emerged dealing with quantum effects on graphs with properties superiour to the corresponding classical operations. For an introductory overview and further references, see Kempe (2003). 

The polar groups are, on the other hand, responsible for an induced polar selectivity. Analytes able to form hydrogen bonds like phenols are retarded more strongly with polar-embedded stationary phases than with the corresponding classical RP of an identical carbon content. This is demonstrated in Figure 2.5 for the separation of polyphenolic compounds present in red wine. The retention time of the polyphenolic compound kaempferol with the shielded phase is more than three times longer than with the corresponding RP column of an identical carbon content. The polar... 

G. Casati In the first part of your talk you discussed the longtime propagation of an initially localized quantum packet. Did you check whether in such situations the corresponding classical motion is in dynamically stable regions or instead is chaotic ... 

Indeed, the maximum time t up to which the quantum packet can propagate before its destruction depends on how far one is in semi-classical regions and scales in different ways depending on the nature of the corresponding classical motion. More precisely, we expect t (1 /h)a for dynamically stable systems and t In h for classically chaotic systems. 

The highly excited and reactive dynamics, the details of which have been made accessible by recently developed experimental techniques, are characterized by transitions between classically regular and chaotic regimes. Now molecular spectroscopy has traditionally relied on perturbation expansions to characterize molecular energy spectra, but such expansions may not be valid if the corresponding classical dynamics turns out to be chaotic. This leads us to a reconsideration of such perturbation techniques and provides the starting point for our discussion. From there, we will proceed to discuss the Gutzwiller trace formula, which provides a semiclassical description of classically chaotic systems. 

Below we shall compare this result with that from the corresponding classical trajectory calculations as well as from variational transition-state treatments. 

Where T is the initial phase point of the system, L is the Liouville operator, y(tf)(F) is the canonical distribution function, and Bk(T) and k(T) are the values of the classical properties Bk and iLk when the system is in the classical state T. Much work has been done to determine how the quantum-mechanical functions approach the corresponding classical functions. 

The classical time-correlation function, , does not obey the condition of detailed balance. Computer experiments provide detailed information about classical time-correlation functions. Is there any way to use the classical functions to predict quantum-mechanical time-correlation functions The answer to this question is affirmative. There exist approximations which enable the quantum-time-correlation functions to be predicted from the corresponding classical functions. Let us denote by v fn(t) the classical time-correlation function and by It(r) the one-sided quantum-mechanical correlation function ... 

Very recently 5- and 8-aminoquinolines have been prepared from the respective aryl bromides by microwave-assisted Buchwald-Hartwig couplings in 10 min at 120°C63. Enhanced yields were observed under microwave conditions as compared to the corresponding classical oil-bath heated reactions (Scheme 2.25). 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

Let us consider the dissociation of a diatomic molecule with internuclear distance R, linear momentum P, and reduced mass m, as illustrated in Figure 6.1. The corresponding classical Hamilton function in the dissociative state is... 

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