Is the nature of quantum chaos classical?
    K.N. Yugay, S.D. Tvorogov, Omsk State University, General Physics Department, pr.Mira,55-A 644077 Omsk, RUSSIA Institute of Atmosphere Optics of Russian Academy of Sciences
    Recently discussions about what is a quantum chaos do not abate [1-16]. Some authors call in question the very fact of an existence of the quantum chaos in nature [8]. Mainly reason to this doubt is what the quantum mechanics equations of motion for the wave function or density matrix are linear whereas the dynamical chaos can arise only into nonlinear systems. In this sence the dynamical chaos in quantum systems, i.e. the quantum chaos, cannot exist. However a number of experimental facts allow us to state with confidence that the quantum chaos exists. Evidently this contradiction is connected with what our traditional description of nature is not quite adequate to it.
    Reflecting on this problem one cannot but pay attention to the following:
    i) two regin exist - the pure quantum one (QR) and the pure classical one (CR), where descriptions are essentially differed. The way in which the quantum and classical descriptions are not only two differen levels of those, but it seems to be more something greater; the problem of quantum chaos indicates to it. Since experimental manifestations of quantum chaos exist therefore one cannot ignore the question on the nature of quantum chaos and the description of it.
    ii) It undoubtedly that the intermediate quantum-classical region (QCR) exists between the QR and the CR, which must be possessed of characteristics both the QR and the CR. Since the term "quasiclassics" is connected traditionally with corresponding approximate method in the quatum mechanics we shall call this region as quantum-classical one further. It is evident that the QCR is the region of high excited states of quantum systems.
    Below shall show that quantum and classical problems are not autonomous into the QCR but they are coupled with each other, so that a solution of a quantum problem contains a solution of a corresponding classical problem, but not vice versa.
    A possible dynamical chaos of a nonlinear classical problem has an effect on the quantum problem so that one can say quantum chaos arises from depths of the nonlinear classical mechanics and it is completely described in terms of nonlinear dynamics, for example, instability, bifurcation, strange attractor and so on. We shall show also that the connection between the quantum and classical problems is reflected on a phase of a wave function which having a quite classical meaning is subjected to its classical equation of motion and in the case of its nonlinearity into the system the dynamical chaos is excited.
    One of a splendid example of a role of the wave function phase is a description of dynamical chaos in a long Josephson junction [17-24]. Here the wave function phase (the difference phases on a junction) of a superconducting condensate is subjected to the nonlinear dynamical sine-Gordon equation. The dynamical chaos arising in a long Josephson junction and describing by the sine-Gordon equation is a quantum chaos essentially since the question is about a phenomenon having exceptionally the quantum character. However the quantum chaos is described here precisely by the classical nonlinear equation.
    Below we shall try to show that the description of the quantum chaos in the more general case may be carry out just as in a long Josephson junction in terms of nonlinear classical dynamics equations of motion to wich the wave function phase of a quantum is subjected. In addition the quantum system must be into the QCR, i.e. into high excited states.
    Let us assume that the Hamiltonian of a system have the form
    
    where the operator of the potential energy U(x,t) is
    
    (We examine here an one-dimensional system for the simplicity). Here U0(x) is the nonperturbation potential energy, and f(t) is the time-dependent external force.
    We shall found the solution of the Schrodinger equation
    
    in the form
    
    where
    
    , is the solution of the classical equation of motion, is the certain constant, s(t) is the time-dependent function, the sense of that will be clear later on. We notice that the function A(x,t) is real. (A representation of the phase A(x,t) in the form (5) at was introduced first by Husimi [25]).
    Substituting (4) into Eq.(1) and taking into account (5), we get
    
    
    
    
    
    Here subscripts t, y and denote the partial derivatives with respect to time t and coordinates y, , respectively.
    On the right of Eq.(6) the expressions of both square brackets are equal to zero because of following relations:
    i) of the classical equation of motion
    
    
    
    
    
    where is the same potential, that is into (3), and
    ii) of the expression for the classical Lagrang function L(t)
    
    so that the function
    
    makes a sense of an action integral.
    Into Eq.(6)
    
    By deduction of Eq.(6) we made use of an potential energy expansion in the form
    
    It is obvious that the expansion (11) is correct in the case when a classical trajectory is close to a quantum one. ............