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

Kicked oscillator: a model of Hamiltonian chaos 5/8 1/2 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”. Cantorous Homoclinic tangle

Localization and resonance in quantum chaotic systems 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) Quantum Classical

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

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. Semi-classics of quantum chaotic systems

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

Stationary phase approximation

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

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 Phase has been accumulated from the turning point! Again, use stationary phase approximation Maslov index Bohr-Sommerfeld quantisation condition with Maslov index

Feynmann path integral result for the propagator Useful (classical) relations Semiclassical propagator Semiclassical Green’s function Monodromy matrix Gutzwiller trace formula 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

Feynman path integral; integral over all possible paths (not only classically allowed ones).

Useful (classical) relations

The semiclassical propagator Only classical trajectories allowed!

The semiclassical propagator

Caustic Focus Zero’s of D correspond to caustics or focus points.

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

The semiclassical propagator

The semiclassical Green’s function

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

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