Semi-classics for non-integrable systems

Lecture 8 of “Introduction to Quantum Chaos”

Kicked oscillator: a model of Hamiltonian chaos

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Poincare-Birkhoff fixed point theoremHomoclinic tangle: generic chaosTori which survives the onset of chaosin phase space the longest has actiongiven 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

QuantumClassical

Universal and non-universal features of quantum chaotic systems

Universal features of eigenvaluespacing.

Quantum scaring ofthe wavefunction.

Classical phase space of non-integrable system is not motion ond-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 spaceand then use stationary phase approximation.

Semi-classics for integrable systems

Solution valid at classical turningpoint

But breaks down here!

Hence, switch back to position space

Semi-classics for integrable systems

Phase has been accumulatedfrom the turning point!

Again, use stationary phase approximation

Maslov index

Bohr-Sommerfeld quantisation conditionwith 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

Feynmann path integral result for the propagator

Feynmann path integral result for the propagator

Feynmann path integral result for the propagator

Feynman path integral; integral overall possible paths (not only classicallyallowed ones).

Useful (classical) relations

Useful (classical) relations

The semiclassical propagator

Only classical trajectoriesallowed!

The semiclassical propagator

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 ofzero’s of inverse D

The semiclassical propagator

The semiclassical Green’s function

The semiclassical Green’s function

Require in terms of action and notHamilton’s principle function

Evaluating the integral with stationary phase approximation leads to

The semiclassical Green’s function

The semiclassical Green’s function

The semiclassical Green’s function

The semiclassical Green’s function

Finally find

Monodromy matrix

Monodromy matrix

For periodic system monodromy matrix coordinate independent

Gutzwiller trace formula

Gutzwiller trace formula

Only periodic orbits contribute to semi-classicalspectrum!

Gutzwiller trace formula

Gutzwiller trace formula

Gutzwiller trace formula

Semiclassical quantum spectrum given by sum of periodicorbit contributions