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IS there any quantum analogy where a three state (or three body) system shows chaotic dynamics as three body problem in classical mechanics?

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Interesting question. Quantum states evolve through linear, unitary state transition operators, so at the fundamental level I would say no - linearity and unitarity means that neighbourhoods are not pathologically "spaghettified" as in classical chaotic systems. But there has to be some way that chaotic behavior can be observed as an emergent behavior of large quantum systems. – WetSavannaAnimal aka Rod Vance Aug 16 '13 at 13:39
@WetSavannaAnimalakaRodVance: Quantum chaos is generic behavior in quantum mechanics. It doesn't require large particle numbers, and linearity and unitarity don't prevent it. – Ben Crowell Aug 16 '13 at 14:20
Rod is correct. True quantum chaotic behavior does not exist. What does exist is quantum systems that exhibit chaotic behavior in the classical limit. The study of this highly non-trivial limiting behavior is referred to as "quantum chaos". – Johannes Aug 16 '13 at 15:06
@Johannes: Sounds like we're just talking about two possible interpretations of the question. I took it to be a question about quantum chaos. – Ben Crowell Aug 16 '13 at 17:10

Quantum chaos is in some sense generic behavior. This article gives two real-world examples from atomic and solid-state physics that are essentially equivalent to the hydrogen atom but with some modification made to the potential. E.g., one of them is that you get chaotic behavior in high-$n$ states in the hydrogen atom, in a strong magnetic field. These two examples can both be considered to be two-body systems. I assume you're asking about three-body systems because for Newtonian gravity, the two-body system doesn't exhibit chaos. I don't know for sure, but I assume that, e.g., a nearly-ionized hekium atom in a strong magnetic field would exhibit the same chaotic behavior as in the case of hydrogen.

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