Editing Quantum chaos (section)

{{Short description|Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory}}
{{Use American English|date = March 2019}}
{{Quantum mechanics}}
[[File:Quantum Chaos.jpg|right|thumb|Quantum chaos is the field of physics attempting to bridge the theories of [[quantum mechanics]] and [[classical mechanics]]. The figure shows the main ideas running in each direction.]]
'''Quantum chaos''' is a branch of [[physics]] focused on how [[chaos theory|chaotic]] classical [[dynamical systems]] can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and [[classical chaos]]?" The [[correspondence principle]] states that classical mechanics is the [[classical limit]] of quantum mechanics, specifically in the limit as the ratio of the [[Planck constant]] to the [[Action (physics)|action]] of the system tends to zero.  If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos).  If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?<ref name="fn_(104)">{{Cite book |last=Haake |first=Fritz |title=Quantum signatures of chaos |date=2001 |publisher=Springer |isbn=978-3-540-67723-9 |edition=2nd rev. and enl. |series=Springer series in synergetics |location=Berlin ; New York}}</ref><ref name="fn_(105)">[[Michael Berry (physicist)|Michael Berry]], "Quantum Chaology", pp 104-5 of ''Quantum: a guide for the perplexed'' by [[Jim Al-Khalili]] ([[Weidenfeld and Nicolson]] 2003), http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf {{Webarchive|url=https://web.archive.org/web/20130308052414/http://www.physics.bristol.ac.uk/people/berry_mv/the_papers/Berry358.pdf |date=2013-03-08 }}.</ref>

In seeking to address the basic question of quantum chaos, several approaches have been employed:
# Development of methods for solving quantum problems where the perturbation cannot be considered small in [[perturbation theory (quantum mechanics)|perturbation theory]] and where quantum numbers are large.
# Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same [[Hamiltonian (quantum mechanics)|Hamiltonian]] (system).
# Study of probability distribution of individual eigenstates (see [[Scar (physics)|scar]]s and [[quantum ergodicity]]).
# [[Semiclassical physics|Semiclassical methods]] such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
# Direct application of the correspondence principle.