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

• 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

Ehrenfest criteria

Ehrenfest criteria

Ehrenfest criteria

Ehrenfest criteria

Ehrenfest criteria

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

Quantum distribution functions: General theory

Quantum distribution functions: General theory

Wigner distribution

This function is not always positive!

Hussimi distribution

Hussimi distribution

Hussimi distribution

Example: Harmonic oscillator

Wave packet centre never followsclassical motion: coherent state neededto describe this. Or….

Example: Kicked rotator

Remarkable resemblance of quantum“phase space” representation of eigenstatewith classical picture.

Example: Kicked rotator

Eigenvalue statistics

Poisson Wigner

Integrable systems

Integrable systems

Uncorrelated eigenvalues

Non-integrable systems

Replace these blocksby random matrices

Non-integrable systems

Symmetry requirements for random matrix blocks

Gaussian ensembles

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

Eigenvalue correlations

Eigenvalue correlations

Eigenvalue correlations

Eigenvalue correlations

Eigenvalue correlations

Eigenvalue correlations

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…?

Eigenvalue correlations