Wave-packet initial motion, spreading, and energy in the periodically kicked pendulum
Boon Leong Lan
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 1994•11 citations