Quantum chaos

As known, classic nonlinear systems can manifest dynamical chaos. On another hand, the fundamental dynamical description must be based on the quantum rather than the classic mechanics. What are the features of quantum systems having the classic analogs with chaotic behavior?

Let us consider a quantum system isolated from the rest world. Its dynamics in Schrodinger representation is regards as a deterministic evolution of the wave function (state vector), and in Heisenberg representation as variation in time of operators associated with dynamical variables (observables). Assuming absence of interaction with macroscopic environment, we exclude dissipation and all the things concerning the measurement theory ("reduction of the state vector"). Hence, following this approach, we deal only with conservative (Hamiltonian) systems.

It seems that simple reasoning leads to a conclusion that chaos is impossible. Indeed, due to the uncertainty principle, in quantum system we cannot speak about presence at a given point of phase space, but only in a cell of finite phase volume determined by the Planck constant. h. Any bounded domain in phase space contains a finite number of the cells. Hence, dynamics regarded as subsequent visiting of the cells cannot be chaotic. Another argument is based on the Schrodinger equation. For finite motion, this equation gives rise to a discrete spectrum of eigenvalues, the allowed energy levels. In accordance with the formula E=hv it means that we have a discrete set of the frequency components. Any possible state is composed as the linear combination of the eigenstates and will evolve quasiperiodically due to the discreteness of the spectrum.

Nevertheless, the classic dynamics must follow from the quantum one as a limit case; this is the celebrated correspondence principle, one of the main fundamental statements of the quantum theory.

Thus, the basic problem of the theory of "quantum chaos" is to explain appearance of chaos in classic limit and to reveal pequliarities of the quantum dynamics associated with classic chaos.

It is worth noting that the problem of the quantization and of passage to classic limit for non-integrable systems was stated and discussed yet in 1917 by Einstein.

As known, for undestanding classic chaos it is proved useful to study artificially constructed special models like Baker's map and Arnold's cat map. It may be thought that the quantum analogs of such systems should be important for understanding the problem of quantum chaos. This idea is elaborated since the pioneering work of Hannay and Berry (1980).

Here we present materials relating to the model quantum system known as Arnold's cat map.

  • Classic Arnold's cat map and other torus maps.
  • What does it mean periodicity in respect to position and momentum in quantum mechanics?
  • Quantum cat map in terms of shift operators. Quantum period.
  • From Heisenberg to Schrodinger.
  • Dynamics of state vector - computer illustrations and analytic consideration.
  • Quantum dynamics in phase space: Husimi distribution and Wigner function.
  • Spectrum of quasienergies
  • Saratov group
    of theoretical nonlinear
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