Physics:Quantum chaos - Revision history

WikiHarold: Remove imported red links from Quantum page

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	~~{{Quantum mechanics}}~~
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	~~[[Category:Chaos theory]]~~
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	~~[[Category:Quantum chaos theory]]~~

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Physics:Quantum chaos\|1}}		{{Sourceattribution\|Physics:Quantum chaos\|1}}

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	~~{{Quantum mechanics topics\|state=expanded}}~~

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	== Berry–Tabor conjecture ==		== Berry–Tabor conjecture ==
	In 1977, Michael Berry and Michael Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.<ref>{{citation \|last=Marklof\|first=Jens \| title=The Berry–Tabor conjecture\|url=http://www.maths.bris.ac.uk/~majm/bib/3ecm.pdf}}</ref><ref>{{cite journal \|author=Barba, J.C.\|title=The Berry–Tabor conjecture for spin chains of Haldane–Shastry type\|journal=EPL\|year=2008\|volume=83\|issue=2 \|article-number=27005 \|doi=10.1209/0295-5075/83/27005\|bibcode = 2008EL.....8327005B \|arxiv = 0804.3685 \|s2cid=53550992 \|display-authors=0}}</ref><ref>{{cite journal\|author=Rudnick, Z.\|title=What Is Quantum Chaos?\|journal=Notices of the AMS\|volume=55\|issue=1\|pages= 32–34\|date=Jan 2008\|url=https://www.ams.org/notices/200801/tx080100032p.pdf}}</ref>		In 1977, Michael Berry and Michael Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.<ref>{{citation \|last=Marklof\|first=Jens \| title=The Berry–Tabor conjecture\|url=http://www.maths.bris.ac.uk/~majm/bib/3ecm.pdf}}</ref><ref>{{cite journal \|author=Barba, J.C.\|title=The Berry–Tabor conjecture for spin chains of Haldane–Shastry type\|journal=EPL\|year=2008\|volume=83\|issue=2 \|article-number=27005 \|doi=10.1209/0295-5075/83/27005\|bibcode = 2008EL.....8327005B \|arxiv = 0804.3685 \|s2cid=53550992 \|display-authors=0}}</ref><ref>{{cite journal\|author=Rudnick, Z.\|title=What Is Quantum Chaos?\|journal=Notices of the AMS\|volume=55\|issue=1\|pages= 32–34\|date=Jan 2008\|url=https://www.ams.org/notices/200801/tx080100032p.pdf}}</ref>

	~~== See also ==~~
	* Scar
	* Statistical mechanics

	== See also ==		== See also ==

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	== Correlating statistical descriptions of quantum mechanics with classical behaviour ==		== Correlating statistical descriptions of quantum mechanics with classical behaviour ==
	[[File:Qdfels.jpg\|400px\|right\|thumb\|Nearest neighbour distribution for Rydberg atom energy level spectra in an electric field as quantum defect is increased from 0.04 (a) to 0.32 (h). The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a Poisson distribution (a) to that of ~~[[Random matrix#Gaussian ensembles\|~~Wigner's surmise]] (h).]]		[[File:Qdfels.jpg\|400px\|right\|thumb\|Nearest neighbour distribution for Rydberg atom energy level spectra in an electric field as quantum defect is increased from 0.04 (a) to 0.32 (h). The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a Poisson distribution (a) to that of Wigner's surmise (h).]]

	Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Wigner's surmise theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.		Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Wigner's surmise theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.

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