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
Schrödinger 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.

Thus we get the equation for the
function in the form

We pay attention here to three
originating moments: 1) Equation (12) is the Schrödinger equation again,
but without an external force. 2) We have the system of two equations of
motion: quantum Eq.(12) and classical Eq.(7). In a general case these equations
make up the system of bound equations, because the coefficient k can be a
function of classical trajectory, . As we show below a connection
between Eqs. (12) and (7) arises in the case, if classical Eq. (7) is
nonlinear. 3) Classical Eq.(7) contains some dissipative term, and so makes sense of a dissipative
coefficient. The arising of dissipation just into the classical equation is
looked quite naturally - a dissipation has the classical character.

Let us assume that is the potential energy of a linear
harmonic oscillator

where is the certain constant. Then
we have

and

where is the natural frequency of the
harmonic oscillator. Equations (15) and (16) represent the corresponding
equations of the quantum and classical linear harmonic oscillators. We see that
Eqs.(15) and (16) are autonomous with respect to each other. Thus in the case
if the classical limit (16) of the corresponding quantum problem (15) is linear
then the solution of the classical and quantum one are not connected with each
other.

Let us assume now that have a form of the potential energy
of the Duffing oscillator

where , and are some constants. For the
potential energy (17) k takes the form

Then we have the following equations
of motion

where

Equation (20) represents the
equation of motion for a nonlinear oscillator. It is seen, that quantum (19)
and classical (20) equations of motion are coupled with each other.

We return to the discussion of
expansion (11). It is seemed obvious, that the classical and quantum
trajektories coexist and close to each other only into the QCR. Into the pure
quantum region QR and into the pure classical one CR these trajectories cannot
coexist: because into the CR a de Broglie wave packet fails quickli in
consequence of dispersion; into the QR the classical trajectory dissappears in
consequence of uncertainty relations. Thus expansion (11) is correct into the
quantum-classical region QCR only, or in other words into the quasiclassical
region. The QCR is became essential just in cases when a classical problem
proves to be nonlinear.

The transition of a particle from
the low states (from the QR) into high excited
states (into the QCR) is

where A(x,t) is defined with the
expression (5). It is easily seen that the probability of this transition

will be depend on the solution of
the classical equation of motion .

Since the classical problem (19) is
nonlinear, then into its, as it is known [26] dynamical chaos can be arisen.
This chaos will lead to nonregularities in the wave function phase A(x,t) and
also in the function , that in turn will lead to
nonregularities of the probabilities of the transition in high excited states,
and also from high excited states into states of the continuous spectrum. In
this way it can be said that the quantum chaos is the dynamical chaos in the
nonlinear classical problem, defining quantum solutions, from the point of view
of the stated here theory.

These investigations are supported
by the Russian Fund of Fundamental Researches (project No. 96-02-19321).

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