Semiclassical Approach to Mesoscopic Systems: Classical by Daniel Waltner

By Daniel Waltner

This quantity describes mesoscopic platforms with classically chaotic dynamics utilizing semiclassical equipment which mix components of classical dynamics and quantum interference results. Experiments and numerical experiences express that Random Matrix idea (RMT) explains actual homes of those platforms good. This was once conjectured greater than 25 years in the past via Bohigas, Giannoni and Schmit for the spectral homes. because then, it's been a problem to appreciate this connection analytically.
The writer deals his readers a clearly-written and up to date therapy of the subjects coated. He extends prior semiclassical methods that taken care of spectral and conductance homes. He indicates that RMT effects can quite often in basic terms be received semiclassically whilst considering classical configurations now not thought of formerly, for instance these containing multiply traversed periodic orbits.

Furthermore, semiclassics is in a position to describing results past RMT. during this context he experiences the impression of a non-zero Ehrenfest time, that is the minimum time wanted for an first and foremost spatially localized wave packet to teach interference. He derives its signature on a number of amounts characterizing mesoscopic structures, e. g. dc and ac conductance, dc conductance variance, n-pair correlation capabilities of scattering matrices and the space within the density of states of Andreev billiards.

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Scr. : J. Phys. : J. Phys. : Semiclassics beyond the diagonal approximation. D. : J. Phys. : Phys. Rev. Lett. : Spinning Particles-Semiclassics and Spectral Statistics, Springer Tracts in Modern Physics vol. : Handbook of Feynman Path Integrals, Springer Tracts in Modern Physics vol. : Microlocal Analysis for Differential Operators. : Proc. Natl. Acad. Sci. : Chaos in Classical and Quantum Mechanics. : Helv. Phys. : Proc. R. Soc. Lond. : J. Phys. A 10, 371 (1977) References 15. 16. 17. 18. 19. 20.

3 We assumed here that the two trajectories possess the same Maslov index: Although this was not yet shown for pairs of open orbits considered here there are several reasons why the Maslov indices can be assumed not to influence properties of chaotic systems showing behaviour predicted by RMT: For special chaotic systems like surfaces of constant negative curvature the Maslov indices do not depend on the specific orbit and thus drop out in Eq. 47). Furthermore the following interpretation is helpful: The Maslov index is given by the number of times the stable and unstable manifolds rotate by half a turn plus twice the number of reflections on walls with Dirichlet boundary 28 2 Semiclassical Techniques applied the classical sum rule with the modification explained before Eq.

28) with β ≡ ±β. An analogous condition is obtained for the momenta p y . Thus only those paths which enter into the cavity at (x , y ) with a fixed angle sin θ = ±βπ/ (kW2 ) and exit the cavity at (x, y) with angle sin θ = ±απ/ (kW1 ) contribute to tα,β (k). There is an intuitive explanation for this condition: The trajectories are those whose transverse wave vectors on entrance and exit match the wave vectors of the modes in the leads. 2 Introduction into Semiclassical Techniques 21 with the reduced actions Sγ α, β, E ≡ Sγ (r, r , E) + ky sin θ − ky sin θ.

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