Review of lecture 5 and 6
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Transcript of Review of lecture 5 and 6
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Review of lecture 5 and 6
•Quantum phase space distributions: Wigner distribution and Hussimi distribution.
•Eigenvalue statistics: Poisson and Wigner level spacing distribution functions; random matrix theory.
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Quantum phenomena
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Quantum phenomena
• So why is there any chaos at all, classical or quantum?• Answer: Classical mechanics is singular limit of quantum limits.
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Ehrenfest criteria
And why it breaks down for quantum chaotic systems…
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Ehrenfest criteria
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Ehrenfest criteria
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Ehrenfest criteria
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Ehrenfest criteria
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Ehrenfest criteria
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Ehrenfest criteria
• Exponentially diverging trajectories changes this sitiuation: for conserving systems then some trajectories must be exponetially converging.
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Quantum distribution functions: General theory
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Quantum distribution functions: General theory
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Wigner distribution
This function is not always positive!
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Hussimi distribution
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Hussimi distribution
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Hussimi distribution
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Example: Harmonic oscillator
Wave packet centre never followsclassical motion: coherent state neededto describe this. Or….
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Example: Kicked rotator
Remarkable resemblance of quantum“phase space” representation of eigenstatewith classical picture.
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Example: Kicked rotator
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Eigenvalue statistics
Poisson Wigner
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Integrable systems
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Integrable systems
Uncorrelated eigenvalues
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Non-integrable systems
Replace these blocksby random matrices
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Non-integrable systems
Symmetry requirements for random matrix blocks
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Gaussian ensembles
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Gaussian ensembles
Thus two classes of random matrix ensembles:
Gaussian Orthogonal EnsembleGaussian Unitary Ensemble
and a third (for case of time reversal + spin ½):
Gaussian Sympleptic Ensemble
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Eigenvalue correlations
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Eigenvalue correlations
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Eigenvalue correlations
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Eigenvalue correlations
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Eigenvalue correlations
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Eigenvalue correlations
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Eigenvalue correlations
All these systems show sameGOE behavior!
Sinai billiardHydrogen atom in strong magnetic fieldNO2 moleculeAcoustic resonance in quartz blockThree dimension chaotic cavityQuarter-stadium shaped plate
Can you match each system to oneof the plots on the right…?
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Eigenvalue correlations