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This is a question that bugging me around for some time now.

It is not clear to me what is the meaning of a chaos if we consider a quantum system.

What is the mathematical formalism (or the quantum analogy) to a orbit that is ergodic in the phase space? Can we still talk of a wave-function describing the particle? I mean, is it still well defined?

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For initial overview I suggest Quantum chaos and Quantum ergodicity on Wikipedia. –  Slaviks Mar 8 '12 at 7:38
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Strictly speaking, there is no quantum chaos. Time evolution is unitary, which implies that small changes in state are not magnified in size. Thus the sensitive dependence on initial conditions, the prerequisite for chaos in classical mechanics, is absent in quantum mechanics. Even stronger, in discrete quantum systems with a finite-dimensional Hilbert space (the case exclusively studied in quantum information theory), dynamics is strictly quasiperiodic.

On the other hand, there is a subject called quantum chaos. However, it does not describe how a quantum system is chaotic, but how to recognize whether a quantum system would become chaotic in the classical limit. (Most studies are numerical only, with very little theoretical support.) This sort of inquiry may be interesting in itself, but has very little relevance to physics, as the classical limit is not really relevant for systems that need a quantum description.

The paper http://mathnt.mat.jhu.edu/zelditch/Preprints/QEM4.pdf by Steve Zelditch surveys theoretical results on a quantum analogue of classical ergodicity, again just related to the question whether the classical limit produces an ergodic system.

Open Problem 4 on p.12 defines a particular class of quantum systems to be ''quantum uniquely ergodic'' if time average and space average differ by a compact operator, whereas classical ergodicity requires them to be equal. But the latter property is crucial to justifying statistical mechanics, where macrocopically, one must average over time, so the system appears homogeneous.

Thus classical ergodic and mixing properties are physically relevant, as they help explain why classical statistical mechanics work. But quantum ergodicity doesn't do the same service for quantum statistical mechanics.


There is a different, more abstract line of research that rephrases ergodicity in operator langauge, which has a physically meaningful quantum analogue even far from the classical regime, though no direct dynamical relevance. This is presented, e.g., in the treatise ''Methods of modern mathematical physics'' by Reed and Simon, in Sections II.5 and VII.4 of Volume I (with interesting Notes p.62 and p.244) and Section XIII.12 of Volume 4 (with Notes p.350ff).

In the classical case, one may write dynamics on phase space $\Omega$ formally in operator form by considering the operators $A(t)$ that maps a phase space function $\psi(z)$ to $\psi(z(t))$, where $z(t)$ is the point reached from $z$ by the classical dynamics in the time interval $[0,t]$. This operator preserves Liouville measure and the positivity of $\psi$. Thus we have a 1-parameter group of operators, hence $A(t)=e^{-tH}$ for some infinitesimal generator $H$ annihilating constant functions. Thus 0 is an eigenvalue of $H$, and by Perron-Frobenius theory, it is the smallest eigenvalue. The dynamics is ergodic iff 0 is a simple eigenvalue. Indeed, $H\psi=0$ iff $U(t)\psi=\psi$ for all $t$ iff $\psi$ is constant on orbits.

This is equivalent to the requirement that $\phi^* A(t)\psi>0$ if $\phi,\psi$ are nonzero and nonnegative and $t$ is sufficiently large. Thus one may (and, essentially, Reed/Simon do) call a 1-parameter group $A(t)$ ergodic if this property holds.

This has an analogure in the quantum case (Reed/Simon, Theorem XIII.44). The place of $H$ is taken by the Hamiltonian, now acting on configuration space functions only, and for a large class of such Hamiltonians, $A(t)=e^{-tH}$ (without the customary $i$, i.e., corresponding to ''imaginary time'' $t$) is positivity preserving. They prove that ergodicity is equivalent to the uniqueness of the ground state.

Note that classically, the lack of ergodicity often shows in the existence of an additional conserved variable beyond functions of the energy. This extends to the quantum case in that the existence of such an additional symmetry invalidates the canonical ensemble as the only equilibrium ensemble. To have equilibrium, this additional conserved quantity must also have a fixed value.

These results are also relevant to constructive QFT in 2 dimensions; see the QFT book by Glimm and Jaffe. They also discuss implications of the lack of uniqueness of the ground state. In quantum field theory, the ground state is the vacuum state. If the vacuum state is not unique, there are different phases.

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The quantum analogue of the ergodic property of a classical manyparticle system is the uniqueness of the ground state in the thermodynamic limit.

See any rigorous treatment of statistical mechanics, e.g., in one of the volumes on mathematical physics by Reed and Simon.

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This is a misleading answer, since there are lots of quantum systems with a unique ground state which don't exhibit chaotic behavior (although there definitely is a connection of unique ground states to quantum chaos). This should be explained in more detail. –  Peter Shor Mar 10 '12 at 13:42
    
Please give an example of a quantum system in_the_thermodynamic_limit (i.e., with infinitely many particles) that substantiates your claim. - I'll look up Reed and Simon to give more precise conditions for my statement, but I don't have it here, so this may take a while. –  Arnold Neumaier Mar 11 '12 at 8:32
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Kitaev's toric code. I realize that this probably isn't the kind of system Reed and Simon were talking about, but without giving Reed and Simon's context, this answer is misleading. –  Peter Shor Mar 11 '12 at 11:37
    
@Peter Shor: Ergodicity and chaotic behavior are far from equivalent for a finite state system. Kitaev's toric code is for a finite state system, where the classical analogue would be a hopping process among the finitely many states, which is never chaotic. But it is ergodic iff the hopping map is transitive, and again, the quantum equivalent of this is the uniqueness of the ground state. -- See my second answer for Reed and Simon's context. –  Arnold Neumaier Mar 13 '12 at 16:56
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Chaos in Classical Mechanics means that a system is extremely sensitive to initial conditions. There are a lot of different possible ways a system can evolve into and it would be exactly time reversible after running it as long as you want.

In Quantum mechanics, to my knowledge, it can not possibly be time reversible. Quantum mechanics describes all possible states a quantum system can be in depending upon what potential you start with. If the potential you start with is your "initial" condition, then you would get different possible states that your quantum system can be in. You are right, in QM, it is very unclear as to what chaos would mean!

I have not studied time-dependent Quantum Mechanics in depth, I warn you, but from what I know, if you were to allow a wave function to just sit there and it does not interact with anything, then it will just stay the same way forever!

It depends on what you are modeling and the potentials you are considering. In real life, we know of course that the wave function will definitely interact with some potential lying around and its wave function will alter constantly and when you try to find out something about the quantum system, you are altering it too.

In Classical Chaos, some system can be altered so much that it can be extremely radical to its initial condition, but the thing that is preserved is that its time reversible! Taking a look at a quantum system, it too can be altered so much that would make it extremely radical to its initial state as well. Although you can make a quantum system return to the same state, it is not the same thing as time reversibility. In conclusion, I don't think the definition of Chaos can be applied to QM, I think its reserved to Classical Mechanics.

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-1: Quantum dynamics is indeed time-reversible. Google the spin echo experiment. And while it's not immediately clear what quantum chaos would mean, quantum analogs of classical chaos have been studied (although it is true that they don't behave exactly like classical chaos). –  Peter Shor Mar 9 '12 at 23:02
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