# Does Quantum Chaos exist?

1. Jan 2, 2007

### Quantum River

According to Prof. Sir M. Berry's argument, the Quantum Mechanics transition to Classical Mechanics (when the Planck constant approaches 0) is a singular process. So there is a lot of new physics here.
Let's consider the Schrodinger Equation. If the wave function at t=0 has a small error, could the small error be amplified to a big one in the evolution? Not likely.
But in the Classical Mechanics, according to Henri Poincaré, the small error in the initial conditions will be amplified to a very big one when the time t is big.
According to the Correspondence Principle, the CM is the limit of QM when the Planck constant approaches 0.
So there is some contradiction or inconsistency here.

Exist or Inexist? It is a problem.

2. Jan 2, 2007

### StatMechGuy

I know it exists because I met a few professors/post docs as an undergrad who did work on it. The idea was to start with a wave function, and then give it boundary conditions that led to chaotic dynamics in classical systems, and track how the wave equation time evolves.

As I recall, there was some interesting stuff that arose from this, such as scarring (the wave function seems to remember where it was earlier). However, I'm not very knowledgeable about the detail. I remember the term "quantum billiards" being thrown around a lot, so that might be a good place to start.

3. Jan 2, 2007

### Quantum River

I think the Quantum Chaos has an underlying difficulty, how we should understand the Particle-Wave duality. When we mention the word Chaos, we are actually discussing a particle system (n-body).
Has we discussed Chaos when considering a wave/field (such as electromagnetic field, phonic wave) problem?
But there is another possibility, we are using the word turbulence when discussing the wave.
So is there turbulence in Quantum wave function ?
Is there Chaos in the Bohm's interpretation? If not, Why?

Last edited: Jan 2, 2007
4. Jan 2, 2007

### StatMechGuy

Turbulence does not necessarily equal chaos. There is chaotic dynamics in a lot of physical systems; the chaos is a consequence in classical situations of the sensitivity of the system to inital conditions. I'm not sure what you think the definition of chaos is.

As for turbulence in a quantum wave function, you can actually see something along those lines in superfluid helium 3. However, that's just a quantum-driven macroscopic effect.

5. Jan 2, 2007

### sanman

I see that this is reasonable. If you were to reduce the jiggling/perturbations of the virtual particles towards zero, then you would see the quantum mechanical jitteriness diminish to zero, and then you'd just be left with the classical mechanics of Einstein and Newton. So that makes perfect sense to me.

But how do you reduce the Planck's Constant towards zero? I think that obviously this would be done by artificial confinement (aka. QED cavity confinement)

The more you confine your particle, using a suitably narrow cavity, the more wavelengths of virtual particles you are shutting out, and preventing them from jostling/buffeting your particle.

Conversely, there are supposed to be certain QED cavity geometries that increase the "vacuum pressure", and these would correspond to an artificially increased Planck's constant, imho.

Or if you were to accelerate towards the speed of light in a particular direction, then you would feel the Unruh radiation from the virtual particles hitting you from that direction. That could amount to a skewing of the Planck's constant, since the Unruh radiation would be anisotropic.

I would even wonder if gravity is a small skewing of Planck's constant, due to its curvature of space.

6. Jan 2, 2007

### StatMechGuy

The classical limit of quantum mechanics, taking Planck's constant to be small, has nothing to do with confining particles in a cavity or whatnot. Planck's constant sets a scale for the action of a system, and when you are dealing with a quantum system whose value of the action is much larger than Planck's constant, that is $$S(q, \dot{q})/\hbar \rightarrow \infty$$, then you approach the classical limit.

7. Jan 2, 2007

### sanman

Well, that's the same as saying the Heisenberg Uncertainty and the DeBroglie wavelength is negligeable for larger objects, which makes them look classical and not quantum-ish.

But I'm talking about the causal mechanism for Heisenberg's Uncertainty and DeBroglie wavelength. From a causal mechanism point of view, then this would be the kicks from the virtual particles, which themselves can be blocked by cavity confinement, so that the kicks cannot fit inside the cavity.

8. Jan 2, 2007

### Quantum River

I am still confused about Quantum Chaos. But the basic problem/contradiction is that there is classical Chaos and no quantum Chaos. But we all agree when the Planck constant h approaches 0, the QM is CM. The meaning of Planck constant h approaches 0 is that the action S of the system is much larger than Planck constant h. I may not express myself clearly last time.

Prof. Sir M. Berry argues:
According to the correspondence principle, the classical world should emerge from the quantum world whenever Planck's constant h->0 is mathematically singular (added by Quantum River). This fact shared by many physical theories that are limits of other theories) complicates the reduction to classical mechanics. Particular interest attaches to the situation where the classical orbits are chaotic, that is, unpredictable. Then if the sytem is isolated the corresponding quantum motion (e.g. of a wave-packet) cannot be chaotic: this is the 'quantum suppresion of chaos'. Chaos occurs in the world because quantum systems are not isolated: the limit h->0 is unstable, and the associated quantum interference effects are easily destroyed by the tiny uncontrolled influences from the environment, and chaos returns; that is, 'decoherence' suppresses the quantum suppresssion of chaos. [1]

Everyone should read the paper of Prof. Berry.
[1]: Is the moon there when somebody looks? M. Berry.

9. Jan 6, 2007

### gone gator

i expect there is a little something missing in the arguement from "someone works on it" therefore "it exists."

Carnot not only did work on thermodynamics but worked at much of the basic theory while believing that heat was a fluid.

would you claim caloric exists?

10. Jan 6, 2007

### gone gator

agreed
well, perhaps it is easier to say: we find chaotic dynamcis in many of our best models of physical systems.
and it is defined in terms of the infinite time behavious of infinitestimal uncertainties. and theres the rub: if the state of a system is described by intergers then there are no infinitesimal uncertainties. we run into things that are worse than singular limits.

11. Sep 30, 2007

### diegzumillo

I'm starting to study quantum chaos too, and the first thing that cought my attention was that doesn't seem to be a very precise definition to 'chaos'. (at least not in the way we're used to see in math) as StatMechGuy said, the most accepted definition is that dynamical systems sensitive to initial conditions are said to be chaotic.
that said, i think the study of quantum chaos comes when we ask what happens with a system 'we know' it's classically chaotic, under the view of quantum mechanics.

12. Oct 1, 2007

### Demystifier

13. Oct 1, 2007

### Demystifier

Why not?
Recall the Ehrenfest theorem that asserts that the average particle position (the peak of the wave packet) moves according to the classical equations of motion.