# Is that true that real quantum chaos doesn't exist?

I read several books and papers on quantum chaos, to my understanding they all emphases that the quantum chaos does not really exist because the linearity of the Schrodinger equation. Some works were done on the so-called quantum kicked rotors which was used as a quantum counterpart for the classical chaos. The model is quantum however the way they study the rotors are quasi-classical. It just look like to study a classical but quantized map and come to the conclusion that the chaos is what they called 'quantum chaos'. Does it really make sense? Also, someone study the so-called chaos-assist turnelling based on the classical-quantized map, as we all know, in the classical case, no turnelling occurs, so if we start the motion in any stable orbit in the phase space, it not possible to jump into the chaotic area with external force. But they said the turnelling is possible because the model is quantum. Again, it is so confusing because

1. they use a classical map to study the quantum model
2. the map is classical but they consider it should work for quantum case
3. they called quasi-classical method but apply the quantum characteristic without any reason?
• Quantum chaos is the study of the quantum behavior of classically chaotic systems. It addresses the question of what are the signatures of classical chaos in the spectra and wave functions of quantum systems – Thomas Feb 19 '13 at 18:36
• Also, just because Schrodinger's equation is linear in the wave function doesn't mean that the potential need be linear as you add more particles. – Jerry Schirmer Feb 19 '13 at 18:47
• Thanks for the comment. That's also confusing to me, in the quantum language if the potential is nonlinear so why equation is still claimed as linear? It is always hard for me to distinguish the classical and quantum case – user1285419 Feb 19 '13 at 19:10
• Linear in this case means that the equation is linear with respect to the wave function (in the sense of linear function: en.wikipedia.org/wiki/Linear_function ). Specifically, superposition results in an answer that is the sum of the answers that would be given by the component parts. However, if the potential is understood to depend on the wavefunction, then this no longer holds. Even the simple three particle system is wicked tricky. – KDN Feb 20 '13 at 1:16
• There are systems that exhibit quantum chaos but are classically regular. You can't avoid having different definitions of chaos in the two cases. – JohnS Apr 12 '18 at 21:41

## 2 Answers

I don't feel confident enough to post an aswer to this question but, since I ran out of space in the comments section, here we go. The definitions of chaotic behavior are all related to trajectories, they must be dense, mixing and sensitive to initial conditions. Easy enough to bring this definition to classical physics, but we do have a problem trying to use it in quantum mechanics, don't we? we no longer have a trajectory. It is in this sense that it is said there is no chaos in quantum mechanics. However, from physics we know there is a relation between classical (in which we can define chaos) and quantum (in which we can not define chaos), and this relation is that the classical dynamics is an approximation, the universe really is quantum. This urges the question of whether is there a more fundamental definition of chaos, and an underlying mechanisms in quantum mechanics that result in classical chaos. Quantum chaos is all about this question, studying the quantum equivalent of classical chaotic systems.

I don't know what to comment on the schrodinger's equatio linearity remarks. I've heard them before but never really dig any further.

The problem is that there are different definitions used for wave (quantum) chaos and ray (classical particle) chaos. But to keep things simple, consider the Bunimovich stadium. Billiards (the classical case) show sensitive dependence on initial conditions AND mixing of trajectories leading to an asymptotically uniform distribution of trajectories. The sensitive dependence is not sufficient. In the quantum case you need to look at the asymptotic behavior of Dirichlet eigenfunctions of the Schrodinger (or Laplacian) operator. In the Bunimovich case what you find is that the nodal domains converge to a uniform distribution on the stadium. This is called quantum ergodicity. But there are "scars", which amount to rare concentrations. If there are no concentrations, then it's called quantum unique ergodicity. Another definition involves the statistical distribution of energy levels. In an separable (classically regular) system, a histogram of the level spacings converges to a Poisson distribution. However, if you slightly perturb the boundary, you see a transition from Poisson to Wigner statistics (or one of another universal distributions depending on the underlying symmetry such as time-reversal invariance). This transition has been observed experimentally in a number of systems and is the subject of the famous BGS conjecture. http://www.scholarpedia.org/article/Bohigas-Giannoni-Schmit_conjecture.

As you can see, in these examples the chaos comes from the boundary, so even though the equations may be linear, you can get the ergodicity which is crucial.

Here is a quote from Eric Heller: "Random waves are the paradigm for quantum chaos. This is as close as quantum mechanics can come to chaos."