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Semi-classics for non-integrable systems

Semi-classics for non-integrable systems. Lecture 8 of “Introduction to Quantum Chaos”. Kicked oscillator: a model of Hamiltonian chaos. Cantorous. 1/2. 5/8. Poincare- Birkhoff fixed point theorem Homoclinic tangle: generic chaos Tori which survives the onset of chaos

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Semi-classics for non-integrable systems

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  1. Semi-classics for non-integrable systems Lecture 8 of “Introduction to Quantum Chaos”

  2. Kicked oscillator: a model of Hamiltonian chaos Cantorous 1/2 5/8 Poincare-Birkhoff fixed point theorem Homoclinic tangle: generic chaos Tori which survives the onset of chaos in phase space the longest has action given by the “golden mean”. Homoclinic tangle

  3. Localization and resonance in quantum chaotic systems Classical Quantum Quantum Previous lecture: A system that is classically diffusive can be dynamically localized in the analogous quantum case, e.g., kicked rotator, but also can show quantum resonances (Lecture 4)

  4. Universal and non-universal features of quantum chaotic systems Universal features of eigenvalue spacing. Quantum scaring of the wavefunction.

  5. Semi-classics of quantum chaotic systems Classical phase space of non-integrable system is not motion on d-dimensional torus – whorls and tendrils of topologically mixing phase space. Usual semi-classical approach (as we will see) relies on motion on a torus.

  6. WKB approximation neglect in semi-classical limit Can now integrate to find S and A.

  7. Stationary phase approximation

  8. Semi-classics for integrable systems Fourier transform to obtain wavefunction in momentum space and then use stationary phase approximation. Momentum space Position space

  9. Semi-classics for integrable systems Solution valid at classical turning point But breaks down here! Hence, switch back to position space

  10. Semi-classics for integrable systems Again, use stationary phase approximation Phase has been accumulated from the turning point! Maslov index Bohr-Sommerfeldquantisation condition with Maslov index

  11. Semi-classics where the corresponding classical system is not integrable Road map for semi-classics for non-integrable systems: • Feynmann path integral result for the propagator • Useful (classical) relations • Semiclassical propagator • Semiclassical Green’s function • Monodromy matrix • Gutzwiller trace formula

  12. Feynmann path integral result for the propagator

  13. Feynmann path integral result for the propagator

  14. Feynmann path integral result for the propagator

  15. Feynmann path integral result for the propagator Feynman path integral; integral over all possible paths (not only classically allowed ones).

  16. Useful (classical) relations

  17. Useful (classical) relations

  18. The semiclassical propagator Only classical trajectories allowed!

  19. The semiclassical propagator

  20. The semiclassical propagator Zero’s of D correspond to caustics or focus points. Caustic Focus

  21. The semiclassical propagator Maslov index: equal to number of zero’s of inverse D Example: propagation of Gaussian wave packet

  22. The semiclassical propagator

  23. The semiclassical Green’s function

  24. The semiclassical Green’s function Evaluating the integral with stationary phase approximation leads to Require in terms of action and not Hamilton’s principle function

  25. The semiclassical Green’s function

  26. The semiclassical Green’s function

  27. The semiclassical Green’s function

  28. The semiclassical Green’s function Finally find

  29. Monodromy matrix

  30. Monodromy matrix For periodic system monodromy matrix coordinate independent

  31. Gutzwiller trace formula

  32. Gutzwiller trace formula Only periodic orbits contribute to semi-classical spectrum!

  33. Gutzwiller trace formula

  34. Gutzwiller trace formula

  35. Gutzwiller trace formula Semiclassical quantum spectrum given by sum of periodic orbit contributions

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