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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. Quantum phenomena. Quantum phenomena.
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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.
Quantum phenomena • So why is there any chaos at all, classical or quantum? • Answer: Classical mechanics is singular limit of quantum limits.
Ehrenfest criteria And why it breaks down for quantum chaotic systems…
Ehrenfest criteria • Exponentially diverging trajectories changes this sitiuation: for conserving systems then some trajectories must be exponetially converging.
Wigner distribution This function is not always positive!
Example: Harmonic oscillator Wave packet centre never follows classical motion: coherent state needed to describe this. Or….
Example: Kicked rotator Remarkable resemblance of quantum “phase space” representation of eigenstate with classical picture.
Eigenvalue statistics Wigner Poisson
Integrable systems Uncorrelated eigenvalues
Non-integrable systems Replace these blocks by random matrices
Non-integrable systems Symmetry requirements for random matrix blocks
Gaussian ensembles Thus two classes of random matrix ensembles: Gaussian Orthogonal Ensemble Gaussian Unitary Ensemble and a third (for case of time reversal + spin ½): Gaussian Sympleptic Ensemble
Eigenvalue correlations All these systems show same GOE behavior! Sinai billiard Hydrogen atom in strong magnetic field NO2 molecule Acoustic resonance in quartz block Three dimension chaotic cavity Quarter-stadium shaped plate Can you match each system to one of the plots on the right…?