Semi-classics for non-integrable systems

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