Unpredictability of wave function's evolution in nonintegrable quantum systems

It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations O(T ) ∼ Tα. On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms O(T ) ∼ (log2T )β for logarithmically short time and thus there is an algorithmic ”compressibility” of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations O(T ) needed to predict a state of system for time interval T can be used as universal sign of chaos. Chaos as universal phenomenon exists in various systems (for example, in human society) and from this point of view the general approach to chaos should not be based on particular properties of a system. It is well known [1] that motion of nonintegrable classical systems has all attributes of chaos: complexity, unpredictability and randomness. However in nonintegrable quantum systems the apparent signs of chaos seems to be absent [2] and the main direction of studies in the field of ”quantum chaos” is semiclassical analysis of various quantum ”signatures” of classical chaos [3, 4]. Nevertheless, taking into account the fundamental correspondence principle it is reasonble to suppose that evolution of nonintegrable quantum systems should be also unpredictable. As a basis tool to analyze predictability of dynamics we use a number of elementary computational operations O(T ) needed to determine a state of the system for time interval T . Let us consider firstly how a number of elementary computational operations needed for prediction of system’s evolution depends on time T and accuracy ∆ in classical mechanics. It is well known [1] that a distance ||δx(t)|| between initially close phase space points grows with time as ||δx(t)|| = ||δx(0)||f(t), (1) where f(t) ∼ e for nonintegrable systems and f(t) ∼ t for integrable ones. The exponential growth of inevitable computational errors leads to unpredictability of long time evolution in nonintegrable classical systems. To predict a state (trajectory) of the system by the moment T with accuracy ∆ we must make computations with accuracy at least