1. 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.

2. Quantum phenomena

3. So why is there any chaos at all, classical or quantum? Answer: Classical mechanics is singular limit of quantum limits.

4. Ehrenfest criteria And why it breaks down for quantum chaotic systems…

5. Ehrenfest criteria

10. Exponentially diverging trajectories changes this sitiuation: for conserving systems then some trajectories must be exponetially converging.

11. Quantum distribution functions: General theory

13. Wigner distribution This function is not always positive!

14. Hussimi distribution

17. Example: Harmonic oscillator Wave packet centre never follows classical motion: coherent state needed to describe this. Or….

18. Example: Kicked rotator Remarkable resemblance of quantum “phase space” representation of eigenstate with classical picture.

19. Example: Kicked rotator

20. Eigenvalue statistics Poisson Wigner

21. Integrable systems

22. Uncorrelated eigenvalues

23. Non-integrable systems Replace these blocks by random matrices

24. Non-integrable systems Symmetry requirements for random matrix blocks

25. Gaussian ensembles

26. 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

27. Eigenvalue correlations

33. All these systems show same GOE behavior! Sinai billiard Hydrogen atom in strong magnetic field NO 2 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…?

34. Eigenvalue correlations