Quantum signatures of chaos

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. 

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 

At somewhat lower energies, but still within the interval [Em-i,Em], we encounter a second class of resonances. Fig. 10.11 shows that their imaginary parts are still small, but their real parts show a considerable degree of irregularity. The real parts of these levels are no longer well 

There is no problem with the existence of the Rydberg regime. Its existence is guaranteed on the basis of the elementary independent particle model, an excellent approximation of the helium states in the Rydberg regime where n m. 

Although not yet firmly established with the help of computer calculations, there are strong numerical indications that the Ericson regime actually exists. Numerical and analytical arguments for the existence of the Ericson regime are presented in Section 10.5.1. 

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F. Haake, Quantum Signatures of Chaos, Springer, Berlin, 2000. 

Synergetic Economics By W.-B. Zhang Quantum Signatures of Chaos... 

Through many enduring classic texts, such as Haken s Synergetics and Information and Self-Organization, Gardiner s Handbook of Stochastic Methods, Risken s The Fokker Planck-Equation or Haake s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. 

Kudrolli, A., Sridhar, S., Pandey, A. and Ramaswamy, R. (1994). Signatures of chaos in quantum billiards Microwave experiments, Phys. Rev. E49, R11-R14. 

Apart from the reactivity descriptors discussed so far, an important diagnostic of the quantum signature of classical chaos [63], viz., the uncertainty product is also calculated [28] which is defined as below,... 

Signatures of quantum chaos in open chaotic billiards... 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

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. 

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 ... 

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