Quantum Chaos via the Quantum Action
نویسندگان
چکیده
We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system-harmonic oscillators with anharmonic coupling-which is classically a chaotic system. We compare Poincaré sections obtained from the quantum action with those from the classical action. 1. Introduction. The phenomenon of chaos is quite well understood in classical physics. Its origin in terms of nonlinear dynamics has been investigated thoroughly in theoretical physics and mathematics. If one considers nature at the level of atoms, where the laws of quantum mechanics (Q.M.) hold, chaotic phenomena are by far less well understood than in classical (macroscopic) physics [1, 2, 3, 4, 5]. Many questions are unresolved yet. For example one can ask the following questions: What is characteristic for quantum chaos? Which are its essential features? What is the analogue of classical Lyapunov exponents in Q.M.? What is the analogue of classical Poincaré sections and of classical KAM surfaces in Q.M.? Here we take the point of view that the quantum action [6, 7, 8, 9, 10] is the ideal tool to address those questions. In the following we discuss the concept of the quantum action. In Sect.2 we discuss its physical meaning, the question of existence in the mathematical sense, its symmetry properties and methods of its analytical and numerical computation. In Sect.4 we present analytical results for the quantum action in the limit when temperature goes to zero. In Sect.6 we discuss quantum chaos at hand of the 2-D anharmonic oscillator. Finally, we give an outlook on further use of the quantum action, e.g. in cosmology, in the inflationary scenario, in condensed matter, and in particular in supraconductivity, and in quantum chaos with time-dependent Hamiltonians.
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تاریخ انتشار 2002