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The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior. ==Quantum theory== A heuristic postulate called the correspondence principle was introduced to quantum theory by Niels Bohr: it states that, in effect, some kind of continuity argument should apply to the classical limit of quantum systems as the value of Planck's constant normalized by the action of these systems tends to zero. Often, this is approached through "quasi-classical" techniques (cf. WKB approximation). More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than Planck's constant ''ħ'', so the "deformation parameter" ''ħ/S'' can be effectively taken to be zero. (cf. Weyl quantization.) Thus typically, quantum commutators (equivalently, Moyal brackets) reduce to Poisson brackets, in a group contraction. In quantum mechanics, due to 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. It is less clear 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. It is possible to 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. Conversely, in the less well-known approach presented in 1932 by Koopman and von Neumann, the dynamics of classical mechanics have been formulated in terms of an operatorial formalism in Hilbert space, a formalism used conventionally for quantum mechanics.〔Koopman, B. O., Neumann, J. v., Dynamical systems of continuous spectra, Proc. Natl. Acad. Sci. U.S.A., vol. 18 (1932), no. 3, pp. 255–263 ((full text ))〕〔Danilo Mauro: Topics in Koopman-von Neumann Theory, (arXiv:quant-ph/0301172 ) (2003); 〕 In a crucial paper (1933), Dirac〔Dirac, P.A.M. (1933). "The Lagrangian in quantum mechanics", ''Phys. Z. der Sowjetunion'' 3: 64-71.〕 explained how classical mechanics is an emergent phenomenon of quantum mechanics: destructive interference among paths with non-extremal macroscopic actions ''S'' » ''ħ'' obliterate amplitude contributions in the path integral he introduced, leaving the extremal action ''Sclass'', thus the classical action path as the dominant contribution, an observation further elaborated by Feynman in his 1942 PhD dissertation.〔Feynman, Richard P. (1942). Laurie M. Brown. ed. "The Principle of Least Action in Quantum Mechanics", Ph.D. Dissertation, Princeton University. (World Scientific publishers, with title "Feynman's Thesis: a New Approach to Quantum Theory", 2005.) ISBN 978-981-256-380-4.〕 (Further see quantum decoherence.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Classical limit」の詳細全文を読む スポンサード リンク
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