Quantum chaos, properties

Taking the experimentally measured mass spectrum of hadrons up to 2.5 GeV from the Particle Data Group, Pascalutsa (2003) could show that the hadron level-spacing distribution is remarkably well described by the Wigner surmise for / = 1 (see Fig. 6). This indicates that the fluctuation properties of the hadron spectrum fall into the GOE universality class, and hence hadrons exhibit the quantum chaos phenomenon. One then should be able to describe the statistical properties of hadron spectra using RMT with random Hamiltonians from GOE that are characterized by good time-reversal and rotational symmetry. 

Heller, E.J. (1986). Qualitative properties of eigenfunctions of classically chaotic Hamiltonian systems, in Quantum Chaos and Statistical Nuclear Physics, ed. T.H. Seligman and H. Nishioka (Springer, Berlin). 

On the other hand, we know that some chemical reaction systems, especially when highly excited, exhibit quantum chaotic features [16] that is, statistical properties of eigenenergies and eigenvectors are very similar to those of random matrix systems [17]. We call such systems quantum chaos systems. Researchers have also studied how these quantum chaos systems behave under some external... 

In the absence of deterministic chaos in the time evolution of the wave functions of bounded systems, the focus of quantum chaos research shifted towards the identification of the fingerprints of classical chaos in the properties of -0- The usual procedure is to start with a classically chaotic system, quantize it canonically, and then try to identify those characteristics of V in the semiclassical limit (ft -) 0) that give away the chaoticity of the underlying classically chaotic system. 

Thus far, quantum chaos theory has mainly concentrated on the statistical description of real energy levels and their spacings (see Section 4.1.1). The complex helium resonances with their two-dimensional character require a different approach. It may be based on Ginibre s ensemble of complex random matrices (Ginibre (1965)). The investigation of the statistical properties of the helium resonances in the complex plane is certainly a good subject for future research. 

In the present chapter, we examine some of these issues, drawing examples from current research. We begin by considering strongly interacting, many-electron systems, which are not necessarily in the semiclassical limit and therefore not true examples of quantum chaos , but which share many of its statistical properties, and we move on to highly-excited atoms in external fields, which can be followed to the semiclassical limit and are therefore very good systems for the study of quantum chaos . 

We stress that level spacings alone do not suffice to make statements about quantum chaos integrable systems are known which give Wigner distributions [561, 562]. Simple systems are also known whose classical dynamics is chaotic, but whose nearest level spacings differ markedly from the Wigner distribution [563]. For atoms, such systems can have the property that one series is dominant, i.e. has a much higher density of... 

He, J.-H., Y.-Q. Wan, and L. Xu (2007b). Nano-effects, quantum-like properties in electrospun nanofibers. Chaos, Solitons Fractals 33(l) 26-37. 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. 

Especially in the highly excited semiclassical regime the quantum properties and dynamics of atomic and molecular systems are most naturally discussed within the framework of chaos. Not only does chaos theory help to characterize spectra and wave functions, it also makes specific predictions about the existence of new quantum dynamical regimes and hitherto unknown exotic states. Examples are the discovery of frozen planet states in the helium atom by Richter and Wintgen (1990a) and... 

As discussed in Section IV B, statisticality results from the combined action of individual energy eigenstates, some of which lead to statistical behavior and others of which do not. The spatial correlation function approach64 focuses on properties of individual (stationary as well as nonstationary) quantum states and allows a unique labeling of quantum states as chaotic or regular, in a manner that links directly to both classical chaos as well as the statisticality of observables, such as reaction rates. 

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. 

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