The theoretical interpretation of the selection in a linear channel of the Saffman-Taylor finger of width λ=0.5 involves singularities of the zero-surface-tension analytical solutions in the complex plane. In such a physical situation the complex plane can be mapped onto the liquid region in real space. Experiments are presented showing that by a very local disturbance of the viscous flow the observed finger width is changed. The results are in quantitative agreement with the theory; the flow disturbance induces a modification of the singularities in the complex plane and thus changes the finger-width selection.

We present a picture of the weakly nonlinear time-dependent (blinking) traveling-wave state in the convection of binary mixtures as a propagating, spatially confined solution of coupled Landau-Ginzburg equations. Quantitative agreement with the measured slow oscillation frequency is found. In addition, a number of experimental observations regarding the blinking and confined states can be understood in this picture.

Models of athermal self-avoiding fluid and polymerized vesicles have been investigated by Monte Carlo simulations. There is no evidence for crumpling of polymerized vesicles, and the squared radius of gyration is proportional to the number of monomers on the surface, ${\mathit{R}}^2$∼N. Fluid vesicles, on the other hand, exhibit a crumpled shape and ${\mathit{R}}^2$∼${\mathit{N}}^{0.8}$. We explain our model dynamics for fluid surfaces. The time-dependent mean-squared displacement ${\mathit{r}}^2$(t) of a monomer on a fluid surface is ∼${\mathit{t}}^{0.8}$, whereas ${\mathit{r}}^2$(t)∼ √t for polymerized vesicles. The corresponding configurational correlation times are τ∼N for fluid and τ∼${\mathit{N}}^2$ for polymerized vesicles.

We have numerically investigated the pattern formation on the surface of a computer-modeled expanding polymer gel. We obtained the formation process and time evolution of patterns. The dependence of patterns on the values of parameters entering the model is investigated and a qualitative interpretation of the result obtained is presented.

A linear rate equation describes fragmentation with continuous and discrete mass loss typified by consumption of porous reactive solids and two-phase heterogeneous solids. For a mass-dependent fragmentation rate ${\mathit{x}}^{\mathrm{\alpha }}$ and a continuous-mass-loss rate ε${\mathit{x}}^{\gamma }$,σ=γ-α-1<0 yields a ‘‘recession regime’’ where small particles lose mass continuously without breaking, σ>0 yields a ‘‘fragmentation regime’’ where all particles break, and σ=0 yields scaling for α>0. Shattering for α<0 and σ≥0 is runaway fragmentation producing an infinte number of particles in a finite time, Exact and asymptotic solutions, exponent relations, and connections with static percolation are found.

Models for viscous fingering are presented to show a crossover between the diffusion-limited-aggregation (DLA) fractal and the dense pattern by interplay between nonviscous and viscous forces. The velocity has a characteristic property: If the velocity is less than the critical value ${\mathit{v}}_{\mathit{c}}$, the displaced fluid behaves as a nonviscous flow, and if the velocity is larger than ${\mathit{v}}_{\mathit{c}}$, the displaced fluid flow is viscous. By using the dimensional analysis, a dimensionless parameter is found to govern the crossover between the DLA fractal and the dense pattern. It is shown that the crossover length from the DLA fractal to the dense pattern increases with the applied pressure. At a very low flow rate, the pattern becomes the dense structure. At a high flow rate, the pattern becomes the DLA fractal. The similarity of the result with the experiment of the viscous fingering through porous media is discussed. The model can also be applied to the viscous fingering under the gravitational field.

The formulation of the quantum description of the rotation angle of the plane rotator has been beset by many of the long-standing problems associated with harmonic-oscillator phases. We apply methods recently developed for oscillator phases to the problem of describing a rotation angle by a Hermitian operator. These methods involve use of a finite, but arbitrarily large, state space of dimension 2l+1 that is used to calculate physically measurable quantum properties, such as expectation values, as a function l. Physical results are then recovered in the limit as l tends to infinity. This approach removes the indeterminacies caused by working directly with an infinite-dimensional state space. Our results show that the classical rotation angle observable does have a corresponding Hermitian operator with well-determined and reasonable properties. The existence of this operator provides deeper insight into the quantum-mechanical nature of rotating systems.

The problem of quantum interference in the presence of an environment is considered by two approaches. One treats the problem from the point of view of the trace left by the interfering particle on its environment. The other regards the phase accumulation of the interfering waves as a statistical process, and explains the loss of interference in terms of uncertainty in the relative phase. The equivalence of the two approaches is proven for the general case. The two approaches are applied to dephasing of electron interference by photon modes in coherent and thermal states, and to dephasing by electromagnetic fluctuations in metals.

We discuss some simple harmonic solutions of the wave equation with covariant boundary data on a surface S that travels along a wave front. Some examples are given mainly from diffraction problems. The paraxial approximation is also discussed.

The self-energy contribution to the transition (2${\mathit{s}}_{1/2}$2${\mathit{p}}^6$3${\mathit{p}}_{3/2}$${)}_{\mathit{J}=1\mathrm{}}$→2${\mathit{s}}^2$2${\mathit{p}}^6$ in neonlike ${\mathrm{Yb}}^{60+}$ has been measured in an electron-beam ion trap to be 18.4±0.8 eV. This value is 10% larger than the value obtained from multiconfiguration Dirac-Fock calculations in the commonly used effective-charge approach. The measurement supports earlier suggestions that the effective-charge approach may overcorrect the screening effect on the self-energy corrections for transitions in neonlike ions.

Functional relations between the Hartree-Fock density and energy, and the densities and energies obtained within the approximate electron correlation methods, such as many-body perturbation theory and coupled clusters, are examined.

The initial-capture conditions and subsequent near-vacuum cascade are described for protonium (p¯p) formation in collisions of antiprotons (p¯) with the neutral H atom or with the negative ion ${\mathrm{H}}^{\mathrm{-}}$. The p¯p cascade is simulated by Monte Carlo with the initial states determined by previous classical-trajectory Monte Carlo calculations of the cross sections for p¯ capture. The cascades of p¯p formed in corotating beams of p¯ and ${\mathrm{H}}^{\mathrm{-}}$ or in extremely low-density hydrogen gas are very similar, both yielding ∼97% p¯-p annihilation in the 2p state of p¯p and ∼1% each in the 1s, 3d, and higher-p states; the sensitivity to the strong-interaction parameters is examined. The time for the purely radiative cascade is quite long (≫1 μs) for p¯p atoms formed at the upper end of the 0.4≲${\mathit{E}}_{\mathrm{lab}}$≲1.4 a.u. reaction window for p¯+${\mathrm{H}}^{\mathrm{-}}$ collisions. Possible experimental ramifications are discussed.

The complex resonance eigenvalue can easily be obtained by scaling the internal coordinates of the Hamiltonian by a complex factor since then the resonance eigenfunction ${\mathrm{\Psi }}_{\mathrm{\theta }}$ is square integrable. The back-transformation of ${\mathrm{\Psi }}_{\mathrm{\theta }}$, however, yields the exact eigenfunction of the original Hamiltonian only when ${\mathrm{\Psi }}_{\mathrm{\theta }}$ is given in closed form. We show here that in the case when a basis-set method is invoked, even if the ${\lim }_{\mathit{n}\to \mathrm{\infty }}$${\mathrm{\Psi }}_{\mathrm{\theta }}^{\mathit{n}}$ is the exact solution of the transformed problem, the n→∞ limit of the back-rotated ${\mathrm{\Psi }}_{\mathrm{\theta }}^{\mathit{n}}$ may produce a divergent wave function. Our numerical examples suggest that the quality of approximation in the back-rotated wave function strongly depends on the basis set.

The orbital ${\mathrm{\mu }}_{\mathit{L}}$⋅B and diamagnetic ${\mathit{A}}^2$ interactions of electrons and positrons with an external magnetic field lead to a small quadratic Zeeman shift in the L=0, ${\mathit{m}}_{\mathit{S}}$=±1 states of positronium, which, unlike the ${\mathit{m}}_{\mathit{S}}$=0 states, are not affected by the usual, spin-induced Zeeman shifts. For the n=1 states, this shift would be about 2 kHz in a field of 1000 G. These additional shifts are the same in all spin states for L=0 but do depend on n.

Variational procedures for predicting energy differences of many-electron systems are investigated. Several different calculations for few-electron systems are considered that illustrate the problems encountered when a many-electron system is modeled as a core plus outer electrons. It is shown that sequences of increasingly more accurate calculations for outer correlation may converge yielding wrong transition energies. At the same time, accurate core-polarization calculations overestimate the binding energy, requiring a core-valence correction. For the high-spin, core-excited states of Li, it was found that outer correlation only predicted electron affinities as accurately as full-correlation studies. This observation suggested a prediction of the core-excited ${}_{}{}^4{}_{}{}^{}$P${\mathrm{-}}^4$S transition in ${\mathrm{Be}}^{\mathrm{-}}$, based on observed ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}^0$${\mathrm{-}}^3$P transition energies of the neutral species, predicted electron affinities including only outer correlation, and a core-valence correction, that is shown to be in good agreement with experiment. A similar calculation for ${\mathrm{Mg}}^{\mathrm{-}}$ predicts a wavelength of 2895.1 Å for this transition.

The virial theorem is used to derive an exact hierarchy of coupled equations for the kinetic energy functional T[ρ] of an inhomogeneous system of noninteracting fermions. The hierarchy links the nth functional derivative of T[ρ] with respect to the particle density ρ(r) to the (n+1)th functional derivative and to ρ(r) for n=1,2,. . . . These functional derivatives are the analogs of the so-called ‘‘direct correlation functions’’ in the theory of classical fluids, and the hierarchy obtained resembles the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the distribution functions associated with such fluids. Some general properties of the two- and three-point correlation functions are described, in particular, their relations to the linear and quadratic response functions that describe the response of the inhomogeneous system to a change in the potential. The hierarchy is then used to obtain several equivalent expressions for T[ρ], one of which is an infinite series of successively higher-order terms, of which the nth term represents the contribution from the n-point correlation function and the densities associated with those points. The equations are then examined in the homogeneous (uniform) gas limit and the first and second equations are verified explicitly with the aid of the associated response functions. The analysis indicates that the infinite series expression converges rapidly for the three-dimensional gas; it also reveals that for both the one-dimensional and two-dimensional gas, the series terminates after the first few terms, suggesting that the structure of T[ρ] is much simpler in one and two dimensions than in three dimensions.

A method for locating errors in matrices representing Hermitian operators in Hilbert spaces is discussed. The method is based on invariance properties of the first two moments of the spectral density distribution of these operators with respect to unitary transformations of the basis. In particular, applications to eigenvalue problems of the N-electron Hamiltonians in finite-dimensional spin-adapted Hilbert spaces are studied in detail.

A new approach to the separation of electronic and nuclear motions in the molecular and solid-state quantum mechanics is proposed that accounts for the finite velocity of an electron and its retardation in following the nuclear motions. Such a retarded nuclear coordinate transforms the Hamiltonian of the harmonic oscillator into the linear combination of operators forming the Lie algebra of the dynamical Schrödinger group. The time evolution of these operators defines the local time for nuclear and electronic motions. An electronic pair in the same local time has a lower value of the ground energy and a nonvanishing value of the group velocity and electronic current density.

Energetic heavy-ion-impact-induced Ne KLL Auger spectra have been studied by a high-resolution electron spectrometer. To analyze the complex Auger spectra in which many satellite lines are included stemming from multiply ionized neon states, the measured series of interrelated spectra was used with the help of a properly developed computerized procedure. Energy and intensity of the lines in the spectra were determined, and an identification procedure was carried out for all the vacancy configurations from Li-like states to the F-like states (diagram lines). The results were compared with the corresponding theoretical transition energy and intensity values. Based on the above results it was possible to obtain information on the multiple-ionization process at the impact-parameter region of the K-shell ionization. Thus, we have determined the two-dimensional vacancy configuration distribution (2s,2p vacancies) for all the states attainable by Auger transitions. The experimentally determined vacancy configuration distributions did not show any significant difference from the predictions of the independent-particle model. In addition, no definite tendency was found in the ionization probabilities as a function of the degree of ionization.

We present an implementaion of the Feshbach formalism to calculate energy positions and widths of doubly excited states of Be-like ions. These systems can be treated formally as two-electron problems by using a suitable model potential to describe the effect of the 1${\mathit{s}}^2$ core. In this respect, we discuss the pertinence of using the one-parameter model potential ${\mathit{V}}_{\mathit{m}}$=2${\mathit{r}}^{\mathrm{-}1}$-2${\mathit{r}}^{\mathrm{-}1}$(1+αr)${\mathit{e}}^{\mathrm{-}2\mathrm{\alpha }\mathit{r}}$. Energies and widths for 1${\mathit{s}}^2$4l4l’ ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}^{\mathit{e}}$ and ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$ autoionizing states of ${\mathrm{N}}^{3+}$ and ${\mathrm{Ne}}^{6+}$ are reported.