Editing Classical limit (section)

==Quantum theory==
A [[heuristic]] postulate called the [[correspondence principle]] was introduced to [[Bohr model|quantum theory]] by [[Niels Bohr]]: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the [[Planck constant]] normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf. [[WKB approximation]]).<ref>{{cite book |last1=Landau |first1=L. D. |author1-link=Lev Landau |last2=Lifshitz |first2=E. M. |author2-link=Evgeny Lifshitz |year=1977 |title=Quantum Mechanics: Non-Relativistic Theory |edition=3rd |volume=3 |publisher=[[Pergamon Press]] |isbn=978-0-08-020940-1}}</ref>

More rigorously,<ref>{{cite journal |last1=Hepp |first1=K. |author1-link=Klaus Hepp |year=1974 |title=The classical limit for quantum mechanical correlation functions |url=http://projecteuclid.org/download/pdf_1/euclid.cmp/1103859623 |journal=[[Communications in Mathematical Physics]] |volume=35 |issue= 4|pages=265–277 |bibcode=1974CMaPh..35..265H |doi=10.1007/BF01646348 |s2cid=123034390 }}</ref> the mathematical operation involved in classical limits is a [[group contraction]], approximating physical systems where the relevant action is much larger than the reduced Planck constant {{mvar|ħ}}, so the "deformation parameter" {{mvar|ħ}}/{{mvar|S}} can be effectively taken to be zero (cf. [[Weyl quantization]].) Thus typically, quantum commutators (equivalently, [[Moyal bracket]]s) reduce to [[Poisson bracket]]s,<ref>{{Cite journal |last1=Curtright |first1=T. L. |last2=Zachos |first2=C. K. |year=2012 |title=Quantum Mechanics in Phase Space |journal=[[Asia Pacific Physics Newsletter]] |volume=1 |pages=37–46 |doi=10.1142/S2251158X12000069|arxiv=1104.5269 |s2cid=119230734 }}</ref> in a [[group contraction]].

In [[quantum mechanics]], due to [[Werner Heisenberg|Heisenberg's]] [[uncertainty principle]], an [[electron]] can never be at rest; it must always have a non-zero [[kinetic energy]], a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. The typical [[occupation number]]s involved are huge: a macroscopic harmonic oscillator with {{mvar|ω}}&nbsp;=&nbsp;2&nbsp;Hz, {{mvar|m}}&nbsp;=&nbsp;10&nbsp;g, and maximum [[amplitude]] {{mvar|x}}<sub>0</sub>&nbsp;=&nbsp;10&nbsp;cm, has {{math|''S''&nbsp;≈&nbsp;''E''/''ω''&nbsp;≈ ''mωx''{{su|b=0|p=2}}/2&nbsp;≈ 10<sup>−4</sup>&nbsp;kg·m<sup>2</sup>/s}}&nbsp;=&nbsp;{{mvar|ħn}}, so that {{mvar|n}}&nbsp;≃&nbsp;10<sup>30</sup>. Further see [[Coherent states#The wavefunction of a coherent state|coherent states]]. It is less clear, however, how the classical limit applies to chaotic systems, a field known as [[quantum chaos]].

Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using [[Hilbert space]], and classical mechanics using a representation in [[phase space]]. One can bring the two into a common mathematical framework in various ways. In the [[phase space formulation]] of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them, including the violations of [[Liouville's theorem (Hamiltonian)]] upon quantization.<ref>{{cite journal |last1=Bracken |first1=A. |last2=Wood |first2=J. |year=2006 |title=Semiquantum versus semiclassical mechanics for simple nonlinear systems |journal=[[Physical Review A]] |volume=73 |issue=1 |pages=012104 |arxiv=quant-ph/0511227 |bibcode=2006PhRvA..73a2104B |doi=10.1103/PhysRevA.73.012104 |s2cid=14444752 }}</ref><ref>Conversely, in the lesser-known [[Koopman–von Neumann classical mechanics|approach presented in 1932 by Koopman and von Neumann]], the dynamics of classical mechanics have been formulated in terms of an [[operator (physics)|operational]] formalism in [[Hilbert space]], a formalism used conventionally for quantum mechanics.
*{{cite journal |last1=Koopman |first1=B. O. |author1-link=Bernard Koopman |last2=von Neumann |first2=J. |author2-link=John von Neumann |year=1932 |title=Dynamical Systems of Continuous Spectra |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |volume=18 |issue=3 |pages=255–263 |bibcode=1932PNAS...18..255K |doi=10.1073/pnas.18.3.255 |pmid=16587673 |pmc=1076203|doi-access=free }}
*{{cite arXiv |last1= Mauro |first1= D. |year= 2003 |title= Topics in Koopman-von Neumann Theory |eprint=quant-ph/0301172}}
*{{Cite journal |last1=Bracken |first1=A. J. |year=2003 |title=Quantum mechanics as an approximation to classical mechanics in Hilbert space |journal=[[Journal of Physics A]] |volume=36 |issue=23 |pages=L329–L335 |doi=10.1088/0305-4470/36/23/101|arxiv=quant-ph/0210164 |s2cid=15505801 }}
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In a crucial paper (1933), [[Paul Dirac|Dirac]]<ref>{{cite journal |last=Dirac |first=P.A.M. |author-link=Paul Dirac |year=1933 |title=The Lagrangian in quantum mechanics |url=http://www.ifi.unicamp.br/~cabrera/teaching/aula%2015%202010s1.pdf |journal=[[Physikalische Zeitschrift der Sowjetunion]] |volume=3 |pages=64–72}}</ref> explained how classical mechanics is an [[Emergence#Non-living, physical systems|emergent phenomenon]] of quantum mechanics: [[Destructive interference#Quantum interference|destructive interference]] among paths with non-[[extremal]] macroscopic actions {{mvar|S}}&nbsp;»&nbsp;{{mvar|ħ}} obliterate amplitude contributions in the [[path integral formulation|path integral]] he introduced, leaving the extremal action {{mvar|S}}<sub>class</sub>, thus the classical action path as the dominant contribution, an observation further elaborated by [[Richard Feynman|Feynman]] in his 1942 PhD dissertation.<ref>{{cite thesis |last=Feynman |first=R. P. |author-link=Richard Feynman |year=1942 |title=The Principle of Least Action in Quantum Mechanics |type=Ph.D. Dissertation |publisher=[[Princeton University]]}}
:Reproduced in {{cite book |last=Feynman |first=R. P. |year=2005 |editor-last=Brown |editor-first=L. M. |title=Feynman's Thesis: a New Approach to Quantum Theory |publisher=[[World Scientific]] |isbn=978-981-256-380-4 |url-access=registration |url=https://archive.org/details/feynmansthesisne00feyn_0 }}</ref> (Further see [[quantum decoherence]].)