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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).

Список литературы

Zaslavsky G.M., Chirikov B.V. Stochastic Instability of Nonlinear Oscillations // Usp. Fiz. Nauk. 1971. V.105. N.1. P.3-29.

Chirikov B.V., Izrailev F.M., Shepelaynsky D.L. Dynamical Stochasticity in Classical and Quantum Mechanics // Sov. Sci. Rev., Sect.C. 1981. V.2. P.209-223.

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