Improvement of numerical integration algorithms by means of coordinate transformations

We study the effect of coordinate transformations on numerical integration algorithms and the Richardson extrapolation. Present method is based on Hermitian transformed eigenvalue equations and symmetrical tridiagonal matrices. Received 18 March 2002 / Received in final form 23 May 2002 Published online 24 September 2002     

Generation of exact analytic bound state solutions from solvable non-powerlaw potentials by a transformation method

A transformation method has been applied to the exactly solvable Hulthen problem to generate a hierarchy of exactly solved quantum systems in any chosen dimension. The generated quantum systems are, in general, energy-dependent with a single normalized eigenfunction, as the Hulthen potential is a non-powerlaw potential. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential that behaves like a potential qualitatively similar to the Poschl-Teller potential. A second-order application of the transformation method on the Hulthen system produces another Sturmian quantum system and a different method is given to regroup them into a normal quantum system which resembles the Morse potential. Existence of normalizable eigenfunctions for these systems are found to be dependent on the local and asymptotic behaviour of the transformation function. Received 30 August 2000 and Received in final form 16 March 2001     

Perturbation theory for velocity-dependent potentials

In the presence of a velocity-dependent Kisslinger potential, the partial-wave, time-independent Schr?dinger equation with real boundary conditions is written as an equation for the probability density. The changes in the bound-state energy eigenvalues due to the addition of small perturbations in the local as well as the Kisslinger potentials are determined up to second order in the perturbation. These changes are determined purely in terms of the unperturbed probability density, the perturbing local potential, as well as the Kisslinger perturbing potential and its gradient. The dependence on the gradient of the Kisslinger potential stresses the importance of a diffuse edge in nuclei. Two explicit examples are presented to examine the validity of the perturbation formulas. The first assumes each of the local and velocity-dependent parts of the potential to be a finite square well. In the second example, the velocity-dependent potential takes the form of a harmonic oscillator. In both cases the energy eigenvalues are determined exactly and then by using perturbation theory. The agreement between the exact energy eigenvalues and those obtained by perturbation theory is very satisfactory. Received: 24 May 2002 / Accepted: 15 July 2002 / Published online: 3 December 2002 RID="a" ID="a"e-mail: mij@hu.edu.jo Communicated by V. Vento     

Magnetic field tomography

Neutral atoms may be trapped via the interaction of their magnetic dipole moment with magnetic field gradients. One of the possible schemes is the cloverleaf trap. It is often desirable to have at hand a fast and precise technique for measuring the magnetic field distribution. We use for instantaneous imaging the equipotential lines of the magnetic field a diagnostic tool which is based on spatially resolved observation of the fluorescence emitted by a hot beam of sodium atoms crossing a thin slice of resonant laser light within the magnetic field region to be investigated. The inhomogeneous magnetic field spatially modulates the resonance condition between the Zeeman-shifted hyperfine sublevels and the laser light and therefore the amount of scattered photons. We apply this technique for mapping the field of our cloverleaf trap in three dimensions under various conditions. Received 20 March 2001 and Received in final form 12 May 2001     

Photoionization of atoms in parallel electric and magnetic fields: Cross-section calculations using the Lanczos algorithm and the complex coordinate method in a discrete variable representation

A combination of the complex-coordinate method and the Lanczos recursion scheme is implemented in the discrete variable representation (DVR) to obtain total photoionization cross-sections using an iterative procedure. Applications to photoionization of hydrogen atoms in electric fields and sodium atoms in electric and parallel electric and magnetic fields are presented and discussed. Received 15 May 2000 and Received in final form 4 October 2000     

Stark shifts and fine structure of the Li 3 D states

⁶Li 3²D _{3/2, 5/2} states were studied using a diode laser to first excite the 2P _3/2 state and a dye laser to populate the 3²D _{3/2, 5/2} states. The dye laser was modulated by an electro-optic modulator and intersected an atomic beam that passed through a field free region and subsequently through a uniform electric field. A value of 1084.24±0.20 MHz was found for the 3²D fine structure splitting. The scalar and tensor polarizabilities were determined to be α (3D _3/2 ) = - 3.753±0.015, α ₂ (3D _3/2 ) = 2.893±0.017, α (3D _5/2 ) = - 3.772±0.008 and α ₂ (3D _5/2 ) = 4.058±0.013 MHz/(kV/cm)². Received 26 September 2002 / Received in final form 22 January 2003 Published online 11 March 2003 RID="a" ID="a"e-mail: wlaser@yorku.ca     

Numerical calculation of energies of some excited states in a helium atom

The energies of some excited states with the total angular momentum L=0, 1 and 2. the total spin of two electrons S=0 and 1, and the even and odd parities are precisely calculated directly from the Schrödinger equation where the mass of the helium nucleus is finite. Moreover, we find that the solutions to the equation for the excited states have some more nodes, which can be used to distinguish the states with the same spectral term.     

Wronskian perturbation theory

We develop a perturbation method that generalizes an approach proposed recently to treat velocity-dependent quantum-mechanical models. In order to test the present approach we apply it to some simple trivial and nontrivial examples.     

Anharmonic oscillators energies via artificial perturbation method

A new pseudoperturbative (artificial in nature) methodical proposal [#!ref15!#] is used to solve for Schr?dinger equation with a class of phenomenologically useful and methodically challenging anharmonic oscillator potentials . The effect of the [#!ref4!#,#!ref5!#] Padé approximant on the leading eigenenergy term is studied. Comparison with results from numerical (exact) and several eligible (approximation) methods is made. Received 2 July 1999 and Received in final form 18 November 1999     

Effect of ordering ambiguity in constructing the Schrödinger equation on perturbation theory

This work explores the application of perturbation formalism, developed for isotropic velocity-dependent potentials, to three-dimensional Schr?dinger equations obtained using different orderings of the Hamiltonian. It is found that the formalism is applicable to Schr?dinger equations corresponding to three possible ordering ambiguities. The validity of the derived expressions is verified by considering examples admitting exact solutions. The perturbative results agree quite well with the exactly obtained ones.     

Transmission resonances in magnetic-barrier structures

Quantum transport properties of electrons in simple magnetic-barrier (MB) structures and in finite MB superlattices are investigated in detail. It is shown that there exists a transition of transmission resonances, i.e., from incomplete transmission resonances in simple MB structures consisting of unidentical blocks, to complete transmission resonances in comparatively complex MB structures (, n is the number of barriers). In simple unidentical block arrangements in double- and triple-MB structures we can also obtain complete transmission by properly adjusting parameters of the building blocks according to k_y-value (k_y is the wave vector in y direction). Strong suppression of the transmission and of the conductance is found in MB superlattices which are periodic arrangements of two different blocks. The resonance splitting effect in finite MB superlattices is examined. It is confirmed that the rule (i.e., for n-barrier tunneling the splitting would be (n-1)-fold) obtained in periodic electric superlattices can be extended to periodically arranged MB superlattices of identical blocks through which electrons with tunnel, and it is no longer proper for electrons with k _y <0 to tunnel. Received: 18 August 1997 / Revised: 20 September 1997 / Accepted: 13 October 1997     

Precise - and -factor measurements of Ba+ isotopes

Laser-microwave double and triple resonance experiments were performed on clouds of Ba⁺ ions confined in a Penning ion trap to induce and detect electronic and nuclear spin flip transitions. Collisions with buffer gas molecules in the trap was used to reduce the lifetime of a long lived metastable state of the ions, in which population trapping might occur, and to cool the ions to the ambient temperature. Loss of ions from the trap by collisions were prevented by coupling the magnetron and reduced cyclotron motions by an additional r.f. field at the sum frequency of the two motions. Electronic Zeeman transitions in ¹³⁸Ba⁺ and ¹³⁵Ba⁺ were observed at a full width of about 3 kHz at a transition frequency of 80 GHz. The uncertainty of the line center was . From the magnetic field calibration by the cyclotron resonance of electrons stored in the same trap the g_J-factor for both isotopes could be determined to . From radiofrequency induced transitions of ¹³⁵Ba⁺ the nuclear g-factor could be determined . Both measurements improve earlier results by about one order of magnitude. Received: 9 July 1998 / Accepted: 14 July 1998     

The beryllium atom and beryllium positive ion in strong magnetic fields

The ground and a few excited states of the beryllium atom in external uniform magnetic fields are calculated by means of our 2D mesh Hartree-Fock method for field strengths ranging from zero up to 2.35×10⁹ T. With changing field strength the ground state of the Be atom undergoes three transitions involving four different electronic configurations which belong to three groups with different spin projections S _z = 0, - 1, - 2. For weak fields the ground state configuration arises from the 1s ²2s ², S _z = 0 configuration. With increasing field strength the ground state evolves into the two S _z = - 1 configurations 1s ²2s2p _-1 and 1s ²2p _-13d _-2, followed by the fully spin polarised S _z = - 2 configuration 1s2p _-13d _-24f _-3. The latter configuration forms the ground state of the beryllium atom in the high field regime γ > 4.567. The analogous calculations for the Be ⁺ ion provide the sequence of the three following ground state configurations: 1s^22s and 1s ²2p _-1 (S _z = - 1/2) and 1s2p _-13d _-2 (S _z = - 3/2). Received 2 October 2000 and Received in final form 8 January 2001     

High accuracy Dirac-finite-element (FEM) calculations for H 2 + and Th 2 179 +

A two-dimensional, fully numerical approach to the four-component first-order Dirac-differential-equation utilizing the Finite-Element-Method (FEM) is employed for H ₂ ⁺ and Th ₂ ^{179 +} . Using elliptic-hyperbolic coordinates and further one-dimensional singular transformations, scaling transformations and extrapolation techniques (geometrical over iteration steps and logarithmic over grid points) we achieve for the molecules H ₂ ⁺ and Th ₂ ^{179 +} relative accuracies better than 10^-12 for 1(1/2) g energies. Received 16 February 2001 and Received in final form 12 July 2001     

Polarizabilities of rubidium (9-10)S 1/2 and 8D 3/2, 5/2 states

Polarizabilities of several rubidium states were determined by measuring stark shifts of transitions using an electro-optically modulated laser beam to excite an atomic beam. The voltage required for atoms excited by the laser beam in an electric field to be simultaneously in resonance as atoms excited by a frequency sideband of the laser in a field free region was measured. The scalar α and tensor α ₂ polarizabilities were found to be: α (9 S _1/2 ) = 103.77±0.09, α (10 S _1/2 ) = 272.54±0.16, α (8 D _3/2 ) = 230.68±0.25 and α ₂ (8 D _3/2 ) = 26.55±0.10, α (8 D _5/2 ) = 222.68±0.14 and α ₂ (8 D _5/2 ) = 51.91±0.10 MHz/(kV/cm)². The results are 100 times more accurate than previous measurements and are within 1% of those found theoretically using a Coulomb approximation calculation. Received 7 September 2000 and Received in final form 6 December 2000     

Low energy excitations of a quasi-2D Bose-Einstein condensate

Starting from the Gross-Pitaevskii energy functional of the 3D Bose-Einstein Condensate, we derive approximately the energy functional and the effective coupling constant of the quasi-2D condensate. The evolution of the quasi-2D condensate wave function is studied by a variational method. Low energy excitation spectra for both positive and negative scattering lengths are analyzed. The condition of collapse instability of a quasi-2D Bose gas with attractive particle interaction is also proposed. Received 31 October 2001 / Received in final form 1st March 2002 Published online 28 June 2002     

Alignment-orientation conversion in molecules in an external magnetic field caused by a hyperfine structure

The paper presents a discussion on the problem of alignment-orientation conversion in an excited state of molecules. It is shown that a rather strong alignment-orientation conversion effect in the excited molecular state can be caused by a joint action of an external magnetic field and hyperfine interaction. The orientation thus created is transverse, i.e. perpendicular to the direction of the external magnetic field. The magnitude of this effect is analyzed as dependent on molecular parameters. Received 15 July 1999 and Received in final form 17 November 1999     

Mixing of wavefunctions in rectangular microwave billiards

A set-up is described allowing the automatic registration of wavefunctions of quasi-two-dimensional microwave billiards of arbitrary shape. Tests of the apparatus with rectangular shaped billiards showed that a precision of some percent in the wavefunction amplitudes can be obtained, as far as isolated resonances are considered. For the case of overlapping resonances, however, the measurement yields wavefunctions which are close to a symmetric and an antisymmetric linear combination of the original rectangle eigenfunctions. The cause for this at first sight surprising result is discussed. Received 21 January 2000     

Electrodynamic trap for neutral atoms

An electrodynamic trap is proposed that stores cold neutral atoms or nonpolar molecules in their ground state as well as in excited states by means of the quadratic Stark effect. The trap uses an oscillating hexapole field and a superposed static homogeneous field. The dynamics of an atom in this trap can be described as a harmonic oscillation in a static pseudopotential. Stability criteria and sample parameters for a number of atomic species are given. Received: 7 August 1998 / Received in final form: 7 January 1999