Abstract



Quantum manifestations of classical chaos became today as a topic of very active interest. Several years ago, Gutzwiller demonstrated the importance of periodic orbits in the understanding of quantum chaotic dynamics, and Heller studied the eect of that the unstable periodic orbits (PO) had in the eigenfunctions of chaotic systems, by the accumulation of density of probability around some of them, called scarring. Recent studies expose that the correlation diagram constitute an ecient tool to approach to the understanding of the scarring phenomenon [3]. These studies shows that the transition from chaos to order is induced by decreasing the value of ¯h, and a series of broad avoided crossing marks the border between these two regions; scars then appears as mixing of the pairs of interacting wave functions. In this work we have chosen a model system (HO2) that is given by a two degrees of freedom Hamiltonian in scattering coordinates. This system shows that the motion in the coordinate is very floppy and a important degree of coupling is present. This ensures that chaos sets in at low energies. The potential energy surface have a symmetry along the = 0.5 rad. axis, and the two potential wells ( = 0.80 rad. and = 1.20 rad). The classical dynamics can be followed by taking Poincaré Surface of Section (SOS) along the minimum energy path. These SOS, show ergodicity bands from very low energies, of the order of 2500 cm−1; even so, this dynamics is governed by trajectories that place on invariant tori, appearing two kind of bifurcations, at different energies; a pitchfork bifurcation at 4500 cm−1 and a saddle node bifurcation at 7000 cm−1. In other hand, the first 200 eigenvalues and wave functions have been obtained, and have been classified in three kinds attending to the nodal structure and the distribution of zeros of the Husimi function, in regular states ¿ all the zeros lies on a line ¿ , irregular states ¿ the zeros appear uniformly distributed over the whole available phase space ¿, and scar states, ¿ some zeros separate from the distribution curve of the regular states, and locate on points whose position is determined by the periodic points of the complementary scarring PO. Arranz et al. show that this kind of molecular systems undergoes a transition from chaos to order as ¯h and energy decreases. Marcus related the existence of overlapping avoided crossings (AC), where a strong mixing of the corresponding wave function leads to complex nodal pattern (quantum chaos). On the other hand, sharp isolated ACs, where a exchange of character occurs, are typical of the regular case. When examined closely the correlation diagram there exist a series of isolated ACs (marked with symbol in Fig. 1) that can be taken to represent the frontier between chaos and order. At the right side of the mentioned figure, it shows the corresponding wave functions of the several states that forms the avoided crossing in the frontier, states 41, 43 and 45 in the interval of values of ¯h[0.8, 1.0]. The mixing character on the AC at the frontier produce scar states. This process involves three states, and the scar formation takes place by two steps.  
International

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Entity

Universidades Politécnica y Autónoma de Madrid 
Entity Nationality

ESPAÑA 
Place

Madrid (España) 