Статья: Is the nature of quantum chaos classical?

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