Semiclassical dynamics of chaotic motion: Unexpected long-time accuracy

Abstract

Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green's function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell ih). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.PACS numbers: 03.65. Sq, 03.40.Kf, 05.45.+b Semiclassical mechanics has a long, illustrious history of providing both physical insight and approximations to a wide array of quantum-mechanical problems. Being a wave mechanics based solely on input of a classical nature, it focuses all the attention on the properties of a quantum system's classical analog. If the underlying classical dynamics are integrable, the theory is fairly well understood. On the other hand, if the underlying dynamics are chaotic, a number of basic difficulties exist. Indeed, for this situation it was recognized many years ago during the period of the "old quantum theory" that essential complications arose in attempting to approximate stationary quantum solutions [1]. Recently, a great deal of research has been directed toward resolving these problems and yet surprisingly, prior to this work ih^ fundamental semiclassical approximation (by which we mean the semiclassical Green's function in the time domain) has never truly been tested for a chaotic system-its validity is quite unknown. One reason for this gap in understanding is a deep rooted, intuitive pessimism about the approximation's applicability since the nonzero size of Planck's constant must be responsible for some kind of smoothing over the intricate complexity that is the essence of chaotic dynamics. This intuition would suggest that phase-space structures on a scale much finer than h cannot be relevant. A typical argument stresses, for example, that it is easy to find a perturbation which, though barely affecting the quantum system, radically alters each of the classical trajectories-the correspondence principle is failing. A further explanation of this gap lies in the technical difficulty of just evaluating the formal semiclassical expression. In this Letter, we shall outline a method that we have developed for perforrning the evaluation and demonstrate the astonishingly quantitative agreement existing between the quantum and semiclassical dynamics for a chaotic system, the stadium billiard. This agreement extends well past the time when classical structure far finer than a quantum cell is put into the semiclassical mechanics.Our starting point is to consider the quantum-mechanical time-dependent Green's function in a coordinate representation. It is denoted G(q,q';r)=<qk -iWt/h q'),where H is a quantum Hamiltonian. The propagation of an initial state 1'(q;0) is then given as ^PCq;/) = J_°°^c?q'G(q,qV)^(...

Inhibition of mixing in chaotic quantum dynamics

“…Hence, the VVG propagator, which puts zero amplitude in the classically forbidden region, becomes inadequate to describe the dynamics of the wave packet after the exit event. We point out that this failure of the VVG propagator is not necessarily inconsistent with the long-time accuracy achieved by Heller and Tomsovic [11] in the case of the stadium billiard. Since the amplitude in the forbidden regions is evidently negligible in that problem [4], it follows that the stretching of the exponential tail and associated phenomena that we observe must not be significant there.…”

Localization properties of groups of eigenstates in chaotic systems

“…Some time later, Tomsovic and Heller [5] showed that even for those long times, for which classical fine structure had developed on a scale much smaller thanh, the semiclassical propagation of the packet can be carried out with good precission, by computing the corresponding correlation function, C scl (t), as a sum of contributions of the homoclinic excursions of the PO. This procedure has nevertheless some drawbacks, since for example the number of orbits that is necessary to include in the calculation to obtain converged results grows dramatically with time.…”

Exact relations between homoclinic and periodic orbit actions in chaotic systems

“…They give rise to different homoclinic orbits in Eq. (27) since 0γ0 = 0100010 0γ 0 = 0110 (29) which do lead to differing quality of approximation. The 10001 orbit, starts from point (3.18110104534044, 3.18110104534044) and maps back into itself to 12 decimal places in a double precision calculation after 5 iterations.…”

“…In the semiclassical regime, properties of such classical orbits are also extremely important. A few cases are given by periodic [23][24][25] and closed orbit sum rules [26][27][28] that determine quantal spectral properties, and homoclinic (heteroclinic) orbit summations [29,30] generating wave packet propagation approximations. The interferences in such semiclassical sum rules are almost exclusively governed by the orbits' classical action functions and Maslov indices [31][32][33], and thus this information takes on greater importance in the context of the asymptotic properties of quantum mechanics.…”