Semi-classics for non- integrable systems Lecture 8 of “Introduction to Quantum Chaos”
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Transcript of Semi-classics for non- integrable systems Lecture 8 of “Introduction to Quantum Chaos”
![Page 1: Semi-classics for non- integrable systems Lecture 8 of “Introduction to Quantum Chaos”](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649e8f5503460f94b93d01/html5/thumbnails/1.jpg)
Semi-classics for non-integrable systems
Lecture 8 of “Introduction to Quantum Chaos”
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Kicked oscillator: a model of Hamiltonian chaos
5/8
1/2
Poincare-Birkhoff fixed point theoremHomoclinic tangle: generic chaosTori which survives the onset of chaosin phase space the longest has actiongiven by the “golden mean”.
Cantorous
Homoclinic tangle
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Localization and resonance in quantum chaotic systems
Previous lecture: A system that is classically diffusive can be dynamically localized in the analogous quantum case, e.g., kicked rotator,
but also can show quantum resonances (Lecture 4)
Quantum
QuantumClassical
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Universal and non-universal features of quantum chaotic systems
Universal features of eigenvaluespacing.
Quantum scaring ofthe wavefunction.
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Classical phase space of non-integrable system is not motion ond-dimensional torus – whorls and tendrils of topologically mixing phase space.
Usual semi-classical approach (as we will see) relies on motion on a torus.
Semi-classics of quantum chaotic systems
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WKB approximation
neglect in semi-classical limit
Can now integrate to find S and A.
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Stationary phase approximation
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Semi-classics for integrable systems
Position space
Momentum space
Fourier transform to obtain wavefunction in momentum spaceand then use stationary phase approximation.
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Semi-classics for integrable systems
Solution valid at classical turningpoint
But breaks down here!
Hence, switch back to position space
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Semi-classics for integrable systems
Phase has been accumulatedfrom the turning point!
Again, use stationary phase approximation
Maslov index
Bohr-Sommerfeld quantisation conditionwith Maslov index
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• Feynmann path integral result for the propagator• Useful (classical) relations• Semiclassical propagator• Semiclassical Green’s function• Monodromy matrix• Gutzwiller trace formula
Semi-classics where the corresponding classical system is not integrable
Road map for semi-classics for non-integrable systems:
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Feynmann path integral result for the propagator
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Feynmann path integral result for the propagator
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Feynmann path integral result for the propagator
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Feynmann path integral result for the propagator
Feynman path integral; integral overall possible paths (not only classicallyallowed ones).
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Useful (classical) relations
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Useful (classical) relations
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The semiclassical propagator
Only classical trajectoriesallowed!
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The semiclassical propagator
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The semiclassical propagator
Caustic
Focus
Zero’s of D correspond to caustics or focus points.
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The semiclassical propagator
Example: propagation of Gaussian wave packet
Maslov index:equal to number ofzero’s of inverse D
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The semiclassical propagator
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The semiclassical Green’s function
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The semiclassical Green’s function
Require in terms of action and notHamilton’s principle function
Evaluating the integral with stationary phase approximation leads to
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The semiclassical Green’s function
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The semiclassical Green’s function
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The semiclassical Green’s function
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The semiclassical Green’s function
Finally find
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Monodromy matrix
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Monodromy matrix
For periodic system monodromy matrix coordinate independent
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Gutzwiller trace formula
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Gutzwiller trace formula
Only periodic orbits contribute to semi-classicalspectrum!
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Gutzwiller trace formula
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Gutzwiller trace formula
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Gutzwiller trace formula
Semiclassical quantum spectrum given by sum of periodicorbit contributions