Is the nature of quantum chaos classical?
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 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
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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