Development of intellectual learning environment for a course in probability theory

During the research work a software module for the training disciplines "Probability Theory and Mathematical Statistics" has been developed. The module is based on the implementation of intelligent systems through fuzzy sets.В ходе научно-исследовательской работы был разработан программный модуль для обучения дисциплинам «Теория вероятности» и «Математическая статистика». Модуль основан на реализации интеллектуальных систем через нечеткие множества

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity ($\tau$) is also dramatic, having the special limiting case of $\tau \to \infty$ at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.Comment: 27 pages, 8 embedded .eps figures; to appear, Annals of Physic

Relativistic dynamical polarizability of hydrogen-like atoms

Using the operator representation of the Dirac Coulomb Green function the analytical method in perturbation theory is employed in obtaining solutions of the Dirac equation for a hydrogen-like atom in a time-dependent electric field. The relativistic dynamical polarizability of hydrogen-like atoms is calculated and analysed.Comment: 15 pages, 3 figures (not included, but hard copies are available upon request

Decoherence of molecular wave packets in an anharmonic potential

The time evolution of anharmonic molecular wave packets is investigated under the influence of the environment consisting of harmonic oscillators. These oscillators represent photon or phonon modes and assumed to be in thermal equilibrium. Our model explicitly incorporates the fact that in the case of a nonequidistant spectrum the rates of the environment induced transitions are different for each transition. The nonunitary time evolution is visualized by the aid of the Wigner function related to the vibrational state of the molecule. The time scale of decoherence is much shorter than that of dissipation, and gives rise to states which are mixtures of localized states along the phase space orbit of the corresponding classical particle. This behavior is to a large extent independent of the coupling strength, the temperature of the environment and also of the initial state.Comment: 7 pages, 4 figure