Resummation of the Divergent Perturbation Series for a Hydrogen Atom in an Electric Field

We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Pade resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. vol. 317, p. 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Pade approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.Comment: 14 pages, RevTeX, 3 tables, 1 figure; numerical accuracy of results enhanced; one section and one appendix added and some minor changes and additions; to appear in phys. rev.

Classical approach in quantum physics

The application of a classical approach to various quantum problems - the secular perturbation approach to quantization of a hydrogen atom in external fields and a helium atom, the adiabatic switching method for calculation of a semiclassical spectrum of hydrogen atom in crossed electric and magnetic fields, a spontaneous decay of excited states of a hydrogen atom, Gutzwiller's approach to Stark problem, long-lived excited states of a helium atom recently discovered with the help of Poincar$\acute{\mathrm{e}}$ section, inelastic transitions in slow and fast electron-atom and ion-atom collisions - is reviewed. Further, a classical representation in quantum theory is discussed. In this representation the quantum states are treating as an ensemble of classical states. This approach opens the way to an accurate description of the initial and final states in classical trajectory Monte Carlo (CTMC) method and a purely classical explanation of tunneling phenomenon. The general aspects of the structure of the semiclassical series such as renormgroup symmetry, criterion of accuracy and so on are reviewed as well. In conclusion, the relation between quantum theory, classical physics and measurement is discussed.Comment: This review paper was rejected from J.Phys.A with referee's comment "The author has made many worthwhile contributions to semiclassical physics, but this article does not meet the standard for a topical review"

Tunneling of quantum rotobreathers

We analyze the quantum properties of a system consisting of two nonlinearly coupled pendula. This non-integrable system exhibits two different symmetries: a permutational symmetry (permutation of the pendula) and another one related to the reversal of the total momentum of the system. Each of these symmetries is responsible for the existence of two kinds of quasi-degenerated states. At sufficiently high energy, pairs of symmetry-related states glue together to form quadruplets. We show that, starting from the anti-continuous limit, particular quadruplets allow us to construct quantum states whose properties are very similar to those of classical rotobreathers. By diagonalizing numerically the quantum Hamiltonian, we investigate their properties and show that such states are able to store the main part of the total energy on one of the pendula. Contrary to the classical situation, the coupling between pendula necessarily introduces a periodic exchange of energy between them with a frequency which is proportional to the energy splitting between quasi-degenerated states related to the permutation symmetry. This splitting may remain very small as the coupling strength increases and is a decreasing function of the pair energy. The energy may be therefore stored in one pendulum during a time period very long as compared to the inverse of the internal rotobreather frequency.Comment: 20 pages, 11 figures, REVTeX4 styl