Sergue Vinitsky

μ-capture and ortho-para transitions in the muonic molecule ppμ

Abstract The capture of a muon by protons at different hydrogen densities is considered. More precise values of γ-factors for ortho- and para-states of the muonic molecule ppμ are obtained: 2γo = 1.009±0.001 and 2γp = 1.143±0.001. Relativistic effects are taken into account in the muonic molecule, and the rate of ortho-para transitions λop = (7.1±1.2) × 104s−1, caused by these effects, is calculated. With this value of λop the μ-capture rate in liquid hydrogen is found to be Λc = (490±10)s−1 that is in agreement with the value Λc = 460±20 s− recently measured by the Saclay-CERN-Bologna collaboration.

Adiabatic representation in the three-body problem with the Coulomb interaction I. The choice of the effective Hamiltonian

A correct dissociation limit for the exchange reaction (a,c)+b to a+(b,c) has been obtained in the adiabatic representation of the three-body problem (a,b,c). As an example, the energy level calculations of mesic molecules pp mu and dd mu , and the system e+e-e+ are presented. Relations between various difficulties of the adiabatic method and the ways of overcoming them are discussed.

The Kantorovich method for high-accuracy calculations of a hydrogen atom in a strong magnetic field: low-lying excited states

A new high-accuracy method for calculating the energy values of low-lying excited states of a hydrogen atom in a strong magnetic field (0 ≤ B ≤ 1013G) is developed based on the Kantorovich approach to parametric eigenvalue problems and using the axial symmetry. The initial two-dimensional spectral problem for the Schrodinger equation is reduced to a spectral parametric problem for a one-dimensional equation and a finite set of ordinary second-order differential equations. The rate of convergence is examined numerically and is illustrated with a set of typical examples. The results are in good agreement with precise calculations by other authors.

Adiabatic hyperspherical representation in barycentric coordinates for helium-like systems

The hyperspherical adiabatic (HSA) representation is introduced in barycentric coordinates for helium-like systems in which isotopic and correlation effects are taken into account exactly. Different methods of forming the HSA basis are considered. The general classification of the HSA basis states is suggested, and examples of correlation diagrams are presented. The complete classification of doubly excited states of two-electron systems is given. Different classification schemes are compared. The convergence of the HSA expansion is investigated numerically. The ground-state energy value of the negative hydrogen ion H- is calculated in the six-channel approximation within an accuracy of 10-5 au.

Uncoupled correlated calculations of helium isoelectronic bound states

An uncoupled correlated variational method for the calculation of helium isoelectronic bound states is proposed. New projective coordinates s = r1 + r2, v = (r12) / (r1 + r2), w = (r1-r2) / (r12) are introduced instead of the conventional ones s = r1 + r2, t = r1-r2, u = r12. All matrix elements of the total Hamiltonian and the weight function are expressed as simple products of three one-dimensional integrals. The variational basis is formed by a set of Laguerre polynomials with a single nonlinear parameter and two sets of Jacobi polynomials for the projective coordinates s,v,w, respectively. It provides a reasonable rate of convergence of the energy, E = E(N), with respect to a number N of the basis components of the eigenvector. The proposed method yields the best available energies for the isoelectronic ground states of the helium atom. New estimations of the isotope helium ground states are also presented.

Modified two-centre continuum wavefunction: application to the dissociative double ionization of H2 by electron impact

The fully differential cross section (FDCS) of the double ionization of the hydrogen molecule by electron impact, with the coincidence detection of the two ejected and the scattered electrons, is determined by the application of a product of two modified two-centre Coulomb continuum (MTCC) wavefunctions describing the two ejected electrons. The MTCC, which fulfils the correct boundary conditions asymptotically up to the order O((kr)−2), is obtained in a closed analytical form by solving the Schrodinger equation of an electron with wave vector k and position vector r in the Coulomb field of two fixed nuclei. After the study of the variation of the (FDCS) for some typical geometries the multiply differential cross section of the (e, 3 − 1e) process is determined and compared to the experimental values.

A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation

Abstract A generalization of the Crank–Nicolson algorithm to higher orders for the time-dependent Schrodinger equation is proposed to improve the accuracy of the time approximation. The implicit difference schemes are obtained in terms of the Magnus expansion for the evolution operator and its further factorization with the help of diagonal Pade approximations. Stability of the schemes and conservation of the approximated solution norm are provided by the fact that the Magnus expansion of the evolution operator preserves its unitarity in any order with respect to a time step τ . As an example, a comparison between the numerical and analytical solutions of a model problem for the oscillator with an explicitly time-dependent frequency was performed for the schemes O ( τ 4 ) and O ( τ 6 ) to demonstrate accuracy, efficiency and adequate convergence of the method.

Symbolic-Numerical Algorithm for Generating Cluster Eigenfunctions: Tunneling of Clusters through Repulsive Barriers

A model for quantum tunnelling of a cluster comprising A identical particles, coupled by oscillator-type potential, through short-range repulsive potential barriers is introduced for the first time in the new symmetrized-coordinate representation and studied within the s-wave approximation. The symbolic-numerical algorithms for calculating the effective potentials of the close-coupling equations in terms of the cluster wave functions and the energy of the barrier quasistationary states are formulated and implemented using the Maple computer algebra system. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, the number of the particles A = 2,3,4, and their symmetry type, is analyzed. It is shown that the resonance behavior of the total transmission coefficient is due to the existence of barrier quasistationary states imbedded in the continuum.

Magnus-factorized method for numerical solving the time-dependent Schrödinger equation

Abstract The method of constructing the high-order stable operator-difference schemes for solving the time-dependent Schrodinger equation are proposed as a generalization of the Crank–Nicolson scheme. The schemes are constructed in terms of the Magnus expansion for the evolution operator. The case of the non-homogeneous equation is also considered.

Adiabatic representation in the three-body problem with the Coulomb interaction. II. The effective two-level approximation

For pt.I see ibid., vol.12, no.4, p.567 (1979). A canonical transformation of an infinite system of differential equations describing the motion of the three-body system in the adiabatic basis is suggested. This transformation allows one to reduce the original problem to the solution of a finite set of differential equations. As an example, a system of two differential equations is constructed which represents the infinite (or finite) system of equations within the accuracy (2M)-2, where M-1=mc/M0 is the ratio of the mass, mc, of the negative charged particle, c, to the reduced mass, M0, of two positively charged particles a and b. The physical meaning of the transformation and the solutions obtained is discussed.