Quantized chaos

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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. 

Given the abovementioned bewildering cornucopia of quantum systems that in one way or another all invoke the notion of chaos, we have to ask the question what exactly is quantum chaos We think that quantum chaos comes in three varieties (I) quantized chaos, (II) semi-quantum chaos and (III) quantum chaos. We refer to these three categories as type I, II and III quantum chaos. The division of quantum chaos into these three types arises naturally if quantum systems are characterized according to whether they do or do not show exponential sensitivity and chaos. The three different types of quantum systems are discussed in Sections 4.1, 4.2 and 4.3, respectively. A short preview of the three different types of quantum chaos follows. 

None of the classically chaotic quantum systems so far investigated in the atomic and molecular physics literature exhibits type III quantum chaos. On the other hand, atomic and molecular physics systems provide excellent examples for quantized chaos, the topic of this section. The attractive feature of the term quantized chaos is that it does not imply anything about what happens to the classical chaos when it is quantized. Usually, especially in bounded time independent quantum systems, classical chaos does not survive the quantization process. The quantized system does not exhibit any instabilities, or sensitivity to initial conditions, e.g. sensitivity to small variations in the wave function at time t = 0. 

Over the past decade quantized chaos has become quite an industry . It has been realized that except for the fleld-free hydrogen atom and related two-body atomic systems, all atoms and molecules, starting with the helium atom, can exhibit chaotic behaviour when treated as classical systems. Although the quantum dynamics of these systems do not show... 

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

Quantized chaos, or quantum chaology (see Section 4.1), is about understanding the quantum spectra and wave functions of classically chaotic systems. The semiclassical method is one of the sharpest tools of quantum chaology. As discussed in Section 4.1.3 the central problem of computing the semiclassical spectrum of a classically chaotic system was solved by Gutzwiller more than 20 years ago. His trace formula (4.1.72) is the basis for all semiclassical work on the quantization of chaotic systems. 

The above arguments show that type II wave chaos is a genuine wave phenomenon in classical wave systems. In the context of quantum mechanics, however, type II quantum chaos is only an approximation. This is because classical walls or dynamic boundaries do not exist in quantum mechanics. The dynamical degrees of freedom of the walls, or boundaries, have to be quantized too, resulting in a higher-dimensional, but purely quantum, system, usually of type I. This fact leads us to a promising... 

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

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