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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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**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 in phase space the longest has action given by the “golden mean”. Homoclinic tangle**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)**Universal and non-universal features of**quantum chaotic systems Universal features of eigenvalue spacing. Quantum scaring of the wavefunction.**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.**WKB approximation**neglect in semi-classical limit Can now integrate to find S and A.**Semi-classics for integrable systems**Fourier transform to obtain wavefunction in momentum space and then use stationary phase approximation. Momentum space Position space**Semi-classics for integrable systems**Solution valid at classical turning point But breaks down here! Hence, switch back to position space**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**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**Feynmann path integral result for the propagator**Feynman path integral; integral over all possible paths (not only classically allowed ones).**The semiclassical propagator**Only classical trajectories allowed!**The semiclassical propagator**Zero’s of D correspond to caustics or focus points. Caustic Focus**The semiclassical propagator**Maslov index: equal to number of zero’s of inverse D Example: propagation of Gaussian wave packet**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**The semiclassical Green’s function**Finally find**Monodromy matrix**For periodic system monodromy matrix coordinate independent**Gutzwiller trace formula**Only periodic orbits contribute to semi-classical spectrum!**Gutzwiller trace formula**Semiclassical quantum spectrum given by sum of periodic orbit contributions

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