en>Leo C Stein: A word had been dropped.

2013-07-10T07:39:54Z

A word had been dropped.

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			In [[quantum chaos]], a branch of [[mathematical physics]], '''quantum ergodicity''' is a property of the [[Quantization (physics)\|quantization]] of [[classical mechanics\|classical mechanical systems]] that are [[chaos theory\|chaotic]] in the sense of exponential sensitivity to initial conditions. Quantum ergodicity states, roughly, that in the high-energy limit, the probability distributions associated to [[energy eigenstates]] of a quantized [[ergodic]] [[Hamiltonian mechanics\|Hamiltonian]] tend to a [[Uniform distribution (continuous)\|uniform distribution]] in the classical [[phase space]]. This is consistent with the intuition that the flows of ergodic systems are equidistributed in phase space. By contrast, classical [[completely integrable system]]s generally have periodic orbits in phase space, and this is exhibited in a variety of ways in the high-energy limit of the eigenstates: typically that some form of concentration or "scarring" occurs in the limit.

			The model case of a Hamiltonian is the [[geodesics as Hamiltonian flows\|geodesic Hamiltonian]] on the [[cotangent bundle]] of a [[compact space\|compact]] [[Riemannian manifold]]. The quantization of the geodesic flow is given by the [[fundamental solution]] of the [[Schrödinger equation]]
			:<math>U_t=\exp(it\sqrt{\Delta})</math>
			where <math>\sqrt{\Delta}</math> is the square root of the [[Laplace-Beltrami operator]]. The '''quantum ergodicity theorem''' of Shnirelman, [[Yves Colin de Verdière]], and Zelditch states that a compact Riemannian manifold whose [[unit tangent bundle]] is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the ''n''th eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as ''n'' → ∞.

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			*[[Eigenstate thermalization hypothesis]]
			*[[Ergodic hypothesis]]

			==References==
			* {{Citation \| last=Zelditch\|first=S\|chapter=Quantum ergodicity and mixing of eigenfunctions\|editor1-last=Françoise \| editor1-first=Jean-Pierre \| editor2-last=Naber \| editor2-first=Gregory L. \| editor3-last=Tsun \| editor3-first=Tsou Sheung \| title=Encyclopedia of mathematical physics. Vol. 1, 2, 3, 4, 5 \| publisher=Academic Press/Elsevier Science, Oxford \| isbn=9780125126601 \| id={{MathSciNet \| id = 2238867}} \| year=2006}}
			* {{Citation \| last=Sunada\|first= T\|chapter=Quantum ergodicity \|title=Trend in Mathematics \|publisher=Birkhauser Verlag, Basel\|year=1997 \|pages=175–196}}

			[[Category:Modular forms]]
			[[Category:Chaos theory]]
			[[Category:Ergodic theory]]
			[[Category:Quantum mechanics]]


			{{math-stub}}

196.46.245.50 at 19:46, 5 July 2012

2012-07-05T19:46:33Z

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en>Leo C Stein: A word had been dropped.

196.46.245.50 at 19:46, 5 July 2012