We study the rise of an infinitely long, two-dimensional gas bubble in a tube. Using a boundary-integral formulation, we demonstrate the existence of a novel form of the surface-tension-induced solvability mechanism originally discovered for the Saffman-Taylor finger. We interpret our findings with the help of a local model caricature of the flow equations.

We propose a new dimension, the exponential dimension, to take proper account of the harmonic lengths that vary inversely to the stage of refinement in the multifractal measure. The need for this dimension arises due to the infinite singularity index which is otherwise obtained with these lengths. We find such a dimension in the Julia set and the circle map. We obtain a phase transition in these sets signaled by a crossover from exponential to generalized dimension. This is due to a switch from the dominance of the harmonic lengths to that of the geometric lengths.

We suggest a novel percolation model where the generated clusters are loopless structures. In this model sites are occupied with probability p and blocked with probability 1-p, but those sites closing loops are considered as blocked sites with probability 1 (‘‘green sites’’). Using scaling analysis and numerical simulations in d=2 we find that the static exponents are the same as for conventional percolation. In contrast, the dynamical exponents such as the diffusion and resistance exponents are significantly different. This result supports the argument that in percolation the dynamical exponents are independent of the static exponents.

A measurement of the hyperfine structure of a stable atomic negative ion has been made using optical probing of microwave transitions. Hyperfine-split Zeeman resonances were observed using state-selective photodetachment to probe the ${}_{}{}^2{}_{}{}^{}\mathrm{P}_{3/2}^{}{}_{}{}^{}$ state of ${}_{}{}^{33}{}_{}{}^{}\mathrm{S}_{}^{\mathrm{-}}{}_{}{}^{}$ ions which were stored in a Penning ion trap. The results for the dipole and quadrupole hyperfine constants are a=91.49(9) MHz and b=26.24(23) MHz. The potential for further application of the technique is discussed.

We examine the role of Berry’s geometrical phase [Proc. R. Soc. London, Ser. A 392, 45 (1984)] in the coherent excitation of atoms. The dynamics of a two-level atom prepared in a superposition of its eigenstates will be modified in a nontrivial fashion by the Berry phase. We extend Berry’s formalism to treat the Liouville equation for the density matrix. This extension allows us to discuss statistical mixtures and to incorporate Berry’s phase within the well-known Bloch-vector description of optical resonance. Furthermore, our density-matrix formalism provides the means of investigating Berry’s phase in open, dissipative systems.

Berry’s [Proc. R. Soc. London, Ser. A 392, 45 (1984)] geometrical phase is discussed in the context of dissipative evolution of an interacting spin system, governed by the stochastic Liouville equation. An analytical treatment is given for a possible ESR experiment on an interacting electron-nucleus system, modulated by two-site jumps. Geometrical phases are shown to be relevant to systems of this type, when their Hamiltonian changes slowly with time. A method for obtaining higher-order corrections to the adiabatic approximation is demonstrated. It is found that if the jumps are slow relative to the rate of change of the Hamiltonian, their effect reduces to familiar line broadening, and the geometrical phases may be observed experimentally. Equations are also set up for a similar ESR experiment on an electron-nucleus system undergoing isotropic rotational diffusion, and a brief discussion of the equations follows.

We consider the spectral problem for lattice Schrödinger operators with polynomial potentials. The eigenfunctions in the discrete spectrum of these operators correspond to the trigonometric moments of the periodic solutions of certain ordinary differential equations. Relying on this observation, the classical theory of moments permits the derivation of exact analytical and numerical bounds to the eigenvalues.

A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.

Several methods for systematically calculating higher-order terms in the Magnus expansion of the time-displacement operator are discussed from a unified point of view. It is shown that a quadratic recursive scheme recently introduced in the average Hamiltonian formalism of NMR spectroscopy can be extended to other cases, including the original multiple commutator approach developed by Magnus [Commun. Pure Appl. Math. 7, 649 (1954)].

One-electron wave functions of half-filled-band closed-shell systems have been decomposed into their orthogonal single-determinant valence-bond (VB) components (antisymmetrized products of orthogonal atomic orbitals). Despite the ordering effect of the Fermi hole among electrons of the same spin, favoring spin-density and charge-density waves, the relative independence between the α and β parts of the wave function results in an overestimation of the weight of multiply ionic structures. Comparison is made with the internally correlated functions for different graphs and Hamiltonians (Hubbard, Pariser-Parr-Pople, and ab initio types). The scaling effect of the correlation does not appear as essentially dependent on the ionicity of the VB forms but exponentially linked to the mean energy of each VB component.

If $C_I^{\mathrm{ex}}$ and $C_I$ are the coefficients of, respectively, the exact and single-determinant wave functions of an extended system on the valence-bond (VB) single determinant ${\Phi }_I$, the scaling effect ${\lambda }_I$=$C_I^{\mathrm{ex}}$/$C_I$ of the internal correlation has been empirically shown to follow approximately the law ${\lambda }_I$=exp(${\mathrm{kE}}_I$) where $E_I$=〈${\Phi }_I$‖H‖${\Phi }_I$〉 [see Lepetit, Oujia, Malrieu, and Maynau, preceding paper, Phys. Rev. A 39, 3274 (1989)]. The theoretical foundation of such a law is derived by using a mixed molecular-orbital valence-bond perturbation expansion. This law suggests a single-parameter approximate wave function, ψ= Σ $C_I$exp(${\mathrm{kE}}_I$)${\Phi }_I$. The calculation of an upper bound of the energy is obtained by optimizing k. The method has been tested for π systems of conjugated hydrocarbons treated through Pariser-Parr-Pople or Hubbard Hamiltonians, giving approximately 90% of the correlation energy, and on a few ab initio valence minimal basis-set calculations of small ${\mathrm{Li}}_{\mathrm{n}}$ clusters, with similar accuracy. The method works as well on open-shell problems and on excited states of both neutral and ionic VB character. This work rationalizes and simplifies (by a reduction to a single parameter) a similar proposal due to Stollhoff and Fulde [Z. Phys. B 26, 257 (1977); 29, 231 (1978); J. Chem. Phys. 73, 4548 (1980)].

Numerically available results on energy levels of doubly excited states of two-electron atoms with Z=1, 2, and 7 are analyzed on the basis of a single Rydberg expression for the pair. Pair quantum defects and other parameters are studied as functions of Z and pair quantum numbers.

In technical applications such as reflection seismics and nondestructive testing, imaging principles are successfully used for the reconstruction of inhomogeneities. In quantum-mechanical inverse problems one usually proceeds by parametrizing the unknown potential with some free parameters, which are fitted to the data. In this paper an imaging method is presented for the inversion of quantum-mechanical data. This method is derived by applying a gradient method to the minimization of the least-squares misfit of the data. This leads to an update of the model for the potential as a continuous function of the space variables in terms of the temporal correlation of two wave fields in this model. The theory is formulated in such a way that these wave fields can be computed using a finite-difference method. With current computers the method can be implemented for spherically symmetric potentials, which might include a spin-orbit interaction.

We consider the collective-field-theoretical description of a many-body quantum system (one dimensional for simplicity). We show how the stochastic phase-space quantization prescriptions can be applied to such a system. We define integration measures for the inner product and prove, constructively, that it is not necessary to employ effective quantities (e.g., effective action, Hamiltonian, etc.) in this scheme, unlike the conventional formulation of collective field theory. We offer intuitive interpretations of these results in terms of particle spreading and minimum uncertainty. A specific, simple example is explicitly worked out.

The mini-max principle is extended to work for the approximations to resonances in the square-integrable function space. The hole-projection (or saddle-point) technique for Feshbach resonances, introduced previously by Chung [Phys. Rev. A 20, 1743 (1979)], is derived from the mini-max principle. Limits of applicability of the method are discussed and its generalization based on the Feshbach-type projector technique is given. The generalized method is applied to the case of the 1s2s2$p^2$P° resonance of ${\mathrm{He}}^{\mathrm{-}}$.

We have measured the cross sections for electron capture into the n=3,L,$M_L$ states of hydrogen for protons colliding with helium atoms at energies between 30 and 80 keV. We used a microwave-resonance, optical detection technique, in which a microwave field drives transitions within the n=3 manifold of the hydrogen atoms, and the change in the Balmer-α light emitted in the decay of the excited atoms is observed. The analysis included cascade feeding of the n=3 states by higher n states. The measured cross sections decrease as the angular momentum L increases, and for fixed L they decrease as ‖$M_L$‖ increases. The 3p and 3d cross sections are roughly inversely proportional to the square of the energy over this range of energies, whereas the 3s cross section peaks near 50 keV.

The dispersion of the atomic giant-dipole resonance in the n=4 shell of Te has been calculated and measured by inelastic-electron-scattering spectroscopy. The calculation, using the method of Zangwill and Soven [Phys. Rev. A 21, 1561 (1980)], agrees with the data at q=0 and, when extended to nonzero q, with data for small q. It fails to match the data when q exceeds the inverse of the radius of the n=4 shell. The nature of the correlations that produce the discrepancy is discussed briefly.

We have employed a two-laser double-resonance method to measure, for the first time, the broadening and shift of the Mg i intercombination line (3s3p ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$–3$s^2$ ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$) and first triplet (3s4s ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$–3s3$p^3$$P_2$) transition due to the presence of helium, neon, argon, krypton, and xenon. Our measurements were performed at pressures below 1 atm. A semiclassical impact theory interpretation of our experimental cross sections provides insight into the nature of the interaction potentials between the two pairs of levels and the noble gases. Our results can be explained in terms of attractive dispersive forces and short-range repulsive forces. A comparison is made between the Mg i broadening and shift cross sections and cross-sectional data of intercombination lines of other alkaline-earth elements and resonance transitions of alkali-metal vapors.

Dielectronic recombination cross sections and rates are estimated for ions of the B isoelectronic sequence, (${\mathrm{C}}^{1+}$, ${\mathrm{O}}^{3+}$, ${\mathrm{Ar}}^{13+}$, and ${\mathrm{Fe}}^{21+}$), in which the 2s→2p excitation is involved during the initial capture. A complete set of rates is then generated for all ions by extrapolation. The rates increase rapidly with nuclear core charge $Z_c$, reaching a maximum at $Z_c$∼10–12. A surprisingly smooth variation within the isonuclear sequence around the B-like ions is observed.

When an energetic ion hits a target consisting of atoms lighter than the projectile, the target configuration seen by the ion along its trajectory may differ significantly from the initial state. This nonlinear effect, which originates in rapidly moving recoil atoms, has been investigated by molecular-dynamics simulation mainly for the simple case where the target is a diatomic molecule. Nonlinear behavior has been distinguished from linear behavior by, respectively, switching on and off the interaction between target particles. Pronounced nonlinear effects have been found in energy loss and scattering angle of the ion; they have been classified roughly as clearing-the-way and blocking-the-way effects, depending on whether nonlinear behavior results in a low or high scattering probability. Effects on stopping power appear small for the small systems investigated. For ion-molecule collisions, the important conclusion is made that binary-scattering behavior need not be indicative of a spectator collision but may, if the ion mass exceeds the target mass, originate in the (nonlinear) clearing-the-way effect. The paper ends with a few results and tentative conclusions on ion-surface scattering with heavy ions.