Editing Quantum chaos (section)

== Approaches ==
[[File:Eps3kekq.jpg|300px|right|thumb|Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of <math>\epsilon = -3.0</math>.<ref name="courtney95a">{{Cite journal |last1=Courtney |first1=Michael |last2=Spellmeyer |first2=Neal |last3=Jiao |first3=Hong |last4=Kleppner |first4=Daniel |date=May 1995 |title=Classical, semiclassical, and quantum dynamics in the lithium Stark system |journal=Physical Review A |language=en |volume=51 |issue=5 |pages=3604–3620 |doi=10.1103/PhysRevA.51.3604 |pmid=9912027 |bibcode=1995PhRvA..51.3604C |issn=1050-2947}}</ref>]]

Questions related to the correspondence principle arise in many different branches of physics, ranging from [[nuclear physics|nuclear]] to [[atomic physics|atomic]], [[molecular physics|molecular]] and [[condensed matter physics|solid-state physics]], and even to [[acoustics]], [[microwave]]s and [[optics]]. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics.<ref>{{cite journal |last1=Yan |first1=Bin |last2=Sinitsyn |first2=Nikolai A. |title=Recovery of Damaged Information and the Out-of-Time-Ordered Correlators |journal=Physical Review Letters |volume=125 |pages=040605 |year=2020 |issue=4 |doi=10.1103/PhysRevLett.125.040605  |pmid=32794812 |arxiv=2003.07267 |bibcode=2020PhRvL.125d0605Y |s2cid=212725801 }}</ref>

Important observations often associated with classically chaotic quantum systems are spectral [[level repulsion]], dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in [[scattering]]). In the semiclassical approach of quantum chaos, phenomena are identified in [[spectroscopy]] by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the [[time evolution]] of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular [[amplitude]] distributions.

Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. [[Dyson series]]). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.