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How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that basically underlies on such a linear equation to describe highly nonlinear chaotic properties of classical systems?

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  • $\begingroup$ Chaos has little to do with "nonlinearity" directly, and more to do with the lack of a complete set of action-angle variables. So in that sense you could think of quantum chaos as the absence of a complete set of operators that specify the eigenbasis of the Hamiltonian exactly as the products of the eigenstates of those operators. $\endgroup$ – webb Jun 23 '14 at 21:53
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    $\begingroup$ Quantum Chaos a whole field of study and not something that can be answered in a stack exchange comment; there long books and papers on the topic. See something like Quantum Chaos: An Introduction by Hans-Jürgen Stöckmann. There are also introductions for more specific applications, like Quantum Chaos and Quantum Dots by Nakamura and Harayama. $\endgroup$ – lnmaurer Jun 25 '14 at 13:53
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    $\begingroup$ @webb, I disagree; any introduction on classical chaos will make the point that nonlinearity is a prerequisite for a (finite dimensional) system to be chaotic. Perhaps that's equivalent to your point (which is certainly true), in which case linearity can't be dismissed either. A lot of the literature on quantum chaos makes the point that the Schrodinger equation is linear -- seemingly at odds with classical chaos. $\endgroup$ – lnmaurer Jun 25 '14 at 13:59
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    $\begingroup$ @Inmaurer, "linear" versus "nonlinear" is irrelevant if you look at Hamiltonian systems as Lie algebras. In this context, classical chaos is the absence in an N-dimensional system to have N quantities $J_i$ which commute with the Hamiltonian in Poisson brackets, viz. $\{H, J_i\} = 0$. That the classical harmonic oscillator is non-chaotic is a consequence of its action-angle variables, not its linearity. $\endgroup$ – webb Jun 25 '14 at 19:29
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    $\begingroup$ @webb. Show me an example of a chaotic, linear, finite-dimensional system, and then I'll concede that linear vs nonlinear is irrelevant. What you're saying about action-angle variables is true but meaningless to a beginner. The way most people are introduced to chaos is through studying nonlinear dynamics (e.g. the Lorenz equations), and the nonlinearity is very relevant. People at that level understand differential equations, not Lie algebras. The question user asked is a common and legitimate one. Heck it's almost identical to the first paragraph of one of the books I mentioned. $\endgroup$ – lnmaurer Jun 25 '14 at 21:48
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The commenters on your question already dealt with the nonlinearity aspect, but the question's aspect that sticks out the most is actually how can any classical system, chaotic or not, be described by quantum mechanics, when they seem to describe two different worlds?

The short answer is "no one really knows", but we do know something. This quantum-classical relationship is often equated with decoherence and you can read a bit about it here, here, here, and here, besides of course Wikipedia.

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