From the linear structure of the Schr ̈odinger equation, it is easy to show that the difference between two wave functions cannot increase. Therefore it is tempting to conclude that a consequence of the linear structure of the equation ruling the evolution law, chaos cannot exist in quantum world.
On the other hand although classical mechanics is typically described by nonlinear equations, it is formally analogous to quantum mechanics in many respects. Indeed, the Liouville equation of classical mechanics affords a linear theory for the evolution of probabilities, at the cost of switching from a finite dimensional phase space to an infinitely dimensional function space, analogously to the quantum mechanics based on the Sch ̈odinger equation. For instance considering two initial conditions of the probability density in the phase space it is possible to show that their difference cannot increase, this in spite of the fact that the existence of chaotic behaviour in classical mechanics is rather common.
I discuss how the presence of classical chaos has nontrivial impact of the be-havior of quantum systems; in particular for: the classical limit as emergent property, and the relevance of the coarse-graining description.
Some References:
* Ford, J., Mantica, G., Ristow, G.H. The Arnold cat: failure of the correspondence principle Physica D 50, 493 (1991);
* Berry, M.V. Chaos and the semiclassical limit of Quantum Mechanics: is the Moon there when somebody looks? In: Russell, R.J. et al (Eds.) Quantum Mechanics: Scientific Perspectives on Divine Action, pp. 41. VaticanObservatory CTNS Publications, (2001);
* Mantica, G. Quantum algorithmic integrability: The metaphor of classical polygonal billiards Phys.Rev. E 61, 6434 (2000);
* Zurek, W.H. Decoherence, einselection, and the quantum origins of the classical Rev. Mod. Phys. 75, 715 (2003);
* Falcioni, M., Vulpiani, A., Mantica, G., Pigolotti, S. Coarse-grained probabilistic automata mimicking chaotic systems Phys. Rev. Lett. 91, 44101 (2003).
