# Quantum chaos

1. Jun 23, 2010

### asemanie

what is quantum chaos? dose it exist realy?
the original definition of classical chaos is a system with exponential sensitivity to initial conditions. however in quantum mechanics because of the linear property of the schrodinger equation, variation on initial condition of the wave function will result in no change in the correlation function , which will result in no chaos in quantum mechanics . Berry's correspondence theorem says that all classical chaotic systems also have quantum chaos , calls quantum chaos the quantum mechanics of classically chaotic systems. for quantum chaos , a variation of the Hamiltonian can be introduced to test the stability of the quantum system to the Hamiltonian

Last edited: Jun 23, 2010
2. Jun 23, 2010

### alxm

Depends on what you mean by the term, I suppose.
If by "quantum chaos" you mean exponential sensitivity to initial conditions, making the system unpredictable under time evolution, then AFAIK 'quantum chaos' in that sense does not exist.

Obviously classical chaos does exist, and since quantum mechanics becomes classical mechanics in the classical limit (e.g. $$\hbar \rightarrow 0$$) you have the question of how that happens. So the term 'quantum chaos' usually refers to the study of that question.

3. Jun 23, 2010

### asemanie

Berry's correspondence theorem?

4. Jun 23, 2010

### Pythagorean

Interesing question. My advisor (who I work with chaos in) is also our department QM teacher.

Are there any straightforward answers about how chaos emerges in the classical limit? Is it related to decoherence at all?

5. Aug 19, 2010

### scienceteacher

Chaos can emerge whenever there is a feedback loop....for example when a hockey puck slides across the ice its speed depends on friction and the amount of friction depends on speed.

I could see a group of particles exhibing chaos with a speed / magnetism feedback loop, however, not a single particle's motion being predicted by shrodingers equation.

The main difference between chaos theory and mainstream physics is that choas uses discrete math to deal with feedback loops instead of differential equations. The solutions using differential calculus never give chaos...however using discrete math applied to the same math equation can give chaos. Becuase chaos relies on quantized math I think it embodies the philosophy of quantum

6. Aug 19, 2010

### alxm

This is just utterly wrong. The difference between chaos theory and 'mainstream physics' (as if chaos theory was outside the mainstream) is that chaos theory is a branch of mathematics, not physics.

Chaos is a property of dynamical systems, which do not have to be discrete at all. Differential equations often give chaos, and some forms of chaos such as strange attractors only occur in continuous systems.

Sheesh, and you call yourself a teacher?

7. Aug 20, 2010

### TheAlkemist

alxm, thanks for this clarification. For a second there I was confused after scienceteacher's post...

8. Sep 23, 2011

### jostpuur

I think that the key to this problem is to understand that classically approximately same initial conditions are not the same thing as quantum mechanically approximately same initial conditions.

For example, numbers 1 and 2 are close to each other, and numbers 1 and 10 are farther away from each other. If I define vectors $a,b,c\in\mathbb{R}^{10}$ like this:

$$a = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)$$
$$b = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0)$$
$$c = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1)$$

Would you believe that vectors $a$ and $b$ are close to each other, and vectors $a$ and $c$ farther away from each other? The L2-distances are the same!

$$\|a - b\|_2 = \sqrt{2}$$

$$\|a - c\|_2 = \sqrt{2}$$

Doesn't this give an answer to the original question?

It could be that classical initial conditions $(q(0),p(0))$ and $(\tilde{q}(0),\tilde{p}(0))$ are almost the same. Then chaos has the consequence that at some $t>0$ the points $(q(t),p(q))$ and $(\tilde{q}(t),\tilde{p}(t))$ are completely different.

Then you find corresponding wave packets $\psi(0)$ and $\tilde{\psi}(0)$, which correspond to the previous classical initial conditions. How could it be, that $\psi(t)$ and $\tilde{\psi}(t)$ could be completely different with some $t>0$, since the time evolution is linear? Well because the initial states never where roughly the same the in linear vector space sense!

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I looked at this question when it was posted in 2010, but I gave a little confused answer back then, which I deleted soon. I could swear I had thought about this already in advance, but it took some time to start recalling this stuff...

9. Dec 7, 2011