review of lecture 5 and 6 quantum phase space distributions: wigner distribution and hussimi...
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So why is there any chaos at all, classical or quantum? Answer: Classical mechanics is singular limit of quantum limits.TRANSCRIPT
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