Chaos manifestations

In the concluding remark of Section 5.2 we asked the question whether the transition from confined chaos to global chaos K = Kc can be seen in an experiment with diatomic molecules. The technical feasibility of such an experiment is discussed in Section 5.4. Here we ask the more modest question whether, and if so, how, the transition to global chaos manifests itself within the framework of the quantum kicked rotor. Since the transition to global chaos is primarily a classical phenomenon, we expect that we have the best chance of seeing any manifestation of this transition in the quantum kicked rotor the more classical we prepare its initial state and control parameters. Thus, we choose a small value... 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom ... 

In Section 10.3 we established that the one-dimensional helium atom is classically chaotic. In Section 10.4.1 we computed its quantum spectrum. We extracted periodic orbit information from the spectrum in Section 10.4.3. But so far the main question has not been addressed How does chaos manifest itself in the helium atom Although this question is still the subject of ongoing research, some preliminary answers are provided in this section. 

How does chaos manifest itself in the helium atom This question, raised at the beginning of this section, can now be answered in the fol-... 

A well-known assessment of the success of software project is the CHAOS Manifest of the Standish Group, http / / standishgroup. cora, but it has also been criticised by a number of sources, many of which can be found by simply Google on standish report . However, there seems to be no doubt about the fact that large software projects have had a relatively poor rate of completion on time, within budget, and to agreed performance criteria, whatever may be valid reasons for this... 

The question of non-classical manifestations is particularly important in view of the chaos that we have seen is present in the classical dynamics of a multimode system, such as a polyatomic molecule, with more than one resonance coupling. Chaotic classical dynamics is expected to introduce its own peculiarities into quantum spectra [29, 77]. In Fl20, we noted that chaotic regions of phase space are readily seen in the classical dynamics corresponding to the spectroscopic Flamiltonian. Flow important are the effects of chaos in the observed spectrum, and in the wavefiinctions of tire molecule In FI2O, there were some states whose wavefiinctions appeared very disordered, in the region of the... 

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

Reichl, L. E. (1992), The Transition to Chaos in Conservative Classical Systems Quantum Manifestations, Springer-Verlag, New York. 

L. R. Reichl, The Transition to Chaos Quantum Manifestations, Springer, Berlin, 1992. 

Here, we also include the contributions related to quantum mechanics The chapter by Takami et al. discusses control of quantum chaos using coarsegrained laser fields, and the contribution of Takahashi and Ikeda deals with tunnehng phenomena involving chaos. Both discuss how chaos in classical behavior manifests itself in the quantum counterpart, and what role it will play in reaction dynamics. 

Chaos does not only wreak havoc in otherwise orderly atomic spectra, it also provides a natural framework, indeed a common language, in which one can discuss such seemingly unrelated systems as, e.g., ballistic electrons in mesoscopic semiconductor structures, the hehum atom, and Rydberg atoms in strong external fields. All these systems have one feature in common their classical counterparts are chaotic. Chaos imprints its presence on their spectra and manifests itself in spectral features which are very similar for all these systems (universahty). 

But chaos is more than a tool. There are as yet unsolved philosophical problems in its wake. While relativity and quantum mechanics necessitated - and in fact originated from - a careful analysis of the concepts of space, time and measurement, chaos, already on the classical level, forces us to re-think the concepts of determinism and predictability. Thus, classical mechanics could not be further removed from the dusty subject it is usually portrayed as. On the contrary it is at the forefront of modern scientific research. Since path integrals provide a link between classical and quantum mechanics, conceptual and philosophical problems with classical mechanics are bound to manifest themselves on the quantum level. We are only at the beginning of a thorough exploration of these questions. But one fact is established already chaos has a profound in-fiuence on the quantum mechanics of atoms and molecules. This book presents some of the most prominent examples. 

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

In Chapter 1 we discussed some concepts of chaos, its manifestations and appUcations on an introductory level from a purely quaUtative point of view. The concepts were introduced ad hoc and in a pictorial manner. We will now turn to a more detailed investigation of chaos in order to prepare the tools and concepts needed for the discussion of chaotic atomic and molecular systems. 

With this section we finish our general survey of chaos in classical and quantum mechanics and turn to a discussion of specific examples of the manifestations of chaos in atomic physics. 

With Fig. 6.5 we estabhshed that the phase space of the classical version of the SSE system contains chaotic regions. But since the SSE system is a manifestly quantum mechanical system, the central question is whether the classical chaos in the SSE system is at all relevant for the quantum dynamics, and, if yes, what are the signatures ... 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

One of the best reasons for studying the one-dimensional model of helium is the search for the quantum mechanical manifestations of classical chaos in the helium atom. Since (10.2.4) contains much of the essential physics of the three-dimensional helium atom, it is a natural starting point for quantum chaos investigations. 

One of the most basic features of the helium spectrum is its organization into an infinite sequence of ionization thresholds. This feature is not the result of intricate computations. It is already apparent on the level of the independent particle model of the helium atom (see Section 10.1). All predictions on the quantum manifestations of chaos have to... 

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