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Review of lecture 5 and 6

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

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  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. Quantum phenomena • 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

  6. Ehrenfest criteria

  7. Ehrenfest criteria

  8. Ehrenfest criteria

  9. Ehrenfest criteria

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

  11. Quantum distribution functions: General theory

  12. Quantum distribution functions: General theory

  13. Wigner distribution This function is not always positive!

  14. Hussimi distribution

  15. Hussimi distribution

  16. 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 Wigner Poisson

  21. Integrable systems

  22. Integrable systems 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. 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

  27. Eigenvalue correlations

  28. Eigenvalue correlations

  29. Eigenvalue correlations

  30. Eigenvalue correlations

  31. Eigenvalue correlations

  32. Eigenvalue correlations

  33. 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…?

  34. Eigenvalue correlations

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