PMID: 15244705Jul 13, 2004Paper

Semiclassical approximations based on complex trajectories

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
A D RibeiroM. Baranger

Abstract

The semiclassical limit of the coherent state propagator <z" absolute value of e(-i H T/Planck's over 2 pi) z'> involves complex classical trajectories of the Hamiltonian H(u,v) = <v absolute value of H u> satisfying u(0) = z' and v(T) = z"*. In this work we study mostly the case z' = z". The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as Planck's over 2 pi goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a ph...Continue Reading

References

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Citations

Feb 15, 2013·The Journal of Chemical Physics·Kirill IgumenshchevOleg Prezhdo
Jun 28, 2011·The Journal of Chemical Physics·Thiago F Viscondi, Marcus A M de Aguiar
May 2, 2008·The Journal of Chemical Physics·Yair GoldfarbDavid J Tannor
Aug 12, 2014·The Journal of Chemical Physics·Jakob Petersen, Kenneth G Kay
Nov 13, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·D G LevkovS M Sibiryakov
Nov 13, 2007·Physical Review Letters·D G LevkovS M Sibiryakov
Sep 21, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Kirill Igumenshchev, Oleg Prezhdo

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