Scattered thoughts on chaos in quantum and classical mechanics

by Prof. Angelo Vulpiani

Wednesday 19 Mar 2025, 16:00 → 17:00 Europe/Rome

2G26

From the linear structure of the Schrodinger equation, it is easy to show

that the difference between two wave functions cannot increase. Therefore

it is tempting to conclude that a consequence of the linear structure of the

equation ruling the evolution law, chaos cannot exist in quantum world.

On the other hand although classical mechanics is typically described by

nonlinear equations, it is formally analogous to quantum mechanics in many

respects. Indeed, the Liouville equation of classical mechanics a ords a linear

theory for the evolution of probabilities, at the cost of switching from

a  finite dimensional phase space to an infinitely dimensional function space,

analogously to the quantum mechanics based on the Schroodinger equation.

For instance considering two initial conditions of the probability density in

the phase space it is possible to show that their difference cannot increase,

this in spite of the fact that the existence of chaotic behaviour in classical

mechanics is rather common.

I discuss how the presence of classical chaos has nontrivial impact of the behavior

of quantum systems; in particular for: the classical limit as emergent

property, and the relevance of the coarse-graining description.

 

Some References:

* Ford, J., Mantica, G., Ristow, G.H. The Arnold cat: failure of the corre-

spondence principle Physica D 50, 493 (1991);

* Berry, M.V. Chaos and the semiclassical limit of Quantum Mechanics: is

the Moon there when somebody looks? In: Russell, R.J. et al (Eds.) Quan-

tum Mechanics: Scienti c Perspectives on Divine Action, pp. 41. Vatican

Observatory CTNS Publications, (2001)

* Mantica, G. Quantum algorithmic integrability: The metaphor of classical

polygonal billiards Phys.Rev. E 61, 6434 (2000);

* Zurek, W.H. Decoherence, einselection, and the quantum origins of the

classical Rev. Mod. Phys. 75, 715 (2003);

* Falcioni, M., Vulpiani, A., Mantica, G., Pigolotti, S. Coarse-grained prob-

abilistic automata mimicking chaotic systems Phys. Rev. Lett. 91, 44101

(2003).