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.

Quantum distribution functions: General theory

Wigner distribution This function is not always positive!

Hussimi distribution

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.

Example: Kicked rotator

Eigenvalue statistics Poisson Wigner

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 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…?

Eigenvalue correlations