Chaos in quantum mechanics

With chaos in classical dynamical systems well established, the question arises whether quantum systems axe able to display exponential sensitivity and chaos. The answer is that most quantum systems do not. Not even if their classical counterparts axe chaotic. We can say that chaos is suppressed on the quantum level. An example of this suppression 

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. 

---

I.Guarneri, Relevance of classical chaos in quantum mechanics the hydrogen atom in a monochromatic field , Physics Reports 154 (1987) 77. 

In order to develop the mind set and methods needed to understand and use the fingerprints of chaos in quantum mechanics, we must set to work. Our journey through chaos in atomic physics begins head-on with a schematic, but physical, example of chaos in Section 1.1. The remaining... 

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. 

---