Energy transition densities driven by time-dependent Hamiltonians

The semiclassical approximation for the energy transition probability density in a recent sequence of papers depends on the full unitary operator that has driven the transition and on the trajectories of the driven classical Hamiltonian. Neither of these is explicitly given for a transition generated by a general Hamiltonian, even if it is time-independent, so that a sudden transition was presumed. The theory is here generalized for arbitrary driving Hamiltonians, by basing it on a compound unitary operator that combines four evolutions, a pair generated by the original Hamiltonian and a pair generated by the driving Hamiltonian. The supporting classical structure is again that of closed compound orbits, but now these are composed of four trajectory segments, corresponding to the quantum evolutions. Notwithstanding the increased complexity, all underlying trajectory segments are then generated by Hamiltonians that are known a priory. The phase space integral for the smooth classical background, with respect to variations of the pair of energies, is preserved from the previous papers. The quantum oscillations of the transition density again result from the stationary phase approximation of a double Fourier integral over the trace of the semiclassical compound unitary operator. Even though it now combines four evolutions instead of two, the phases of the oscillations agree with the previous results if the driven Hamiltonian is known

Local quantum ergodic conjecture

The Quantum Ergodic Conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a delta-function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple observables, in agreement with Shnirelman's theorem, but this putative Wigner function violates several important requirements. Consequently, we transfer the conjecture to the Fourier transform of the Wigner function, that is, the chord function. We show that all the relevant consequences of the usual conjecture require only information contained within a small (Planck) volume around the origin of the phase space of chords: translations in ordinary phase space. Loci of complete orthogonality between a given eigenstate and its nearby translation are quite elusive for the Wigner function, but our local conjecture stipulates that their pattern should be universal for ergodic eigenstates of the same Hamiltonian lying within a classically narrow energy range. Our findings are supported by numerical evidence in a Hamiltonian exhibiting soft chaos. Heavily scarred eigenstates are remarkable counter-examples of the ergodic universal pattern.Comment: 4 figure