Energy-level statistics are studied for a one-dimensional integrable system. Direct evidence is found that ‘‘chaotic,’’ i.e., Gaussian-orthogonal-ensemble, level statistics can occur in such an integrable system.

In this paper we study the time evolution of a quasiperiodically kicked spin system whose Hamiltonian is linear in the SU(2) generators. Since this Hamiltonian preserves SU(2) coherent states under time evolution, there is a close correspondence between the quantum evolution and the classical evolution of the system on the Bloch sphere. Such a system has previously been studied by Milonni, Ackerhalt, and Goggin [Phys. Rev. A 35, 1714 (1987)] and, with two incommensurate driving frequencies, the quantum evolution was characterized by (a) decay of the autocorrelation function of the state vector, (b) broadband power spectrum of observables, (c) ergodicity on the Bloch sphere. We expand upon their work to study the corresponding behavior of the motion on a symplectic phase space and the decay of the autocorrelation function of the state vector. When the forcing strength is such that the motion is localized in phase space, the autocorrelation function shows revivals. When the motion is ergodic the autocorrelation function rapidly decays without revivals. The classical motion however is not chaotic, there is no sensitive dependence on initial conditions. For three incommensurate frequencies, the motion is more delocalized than for two frequencies for a fixed forcing strength.

In this paper we incorporate different strengths of the centrifugal potential into a single partial wave function of the WKB approach in different spatial regions. By a suitable choice of these regions and strengths, the WKB wave functions can be made bounded and smooth everywhere as well as compatible with quantum mechanics both close to and far from the origin. This modified WKB approximation becomes exact in the strong-coupling limit. Moreover, by iteration, it develops an absolute convergent series expansion of the correct scattering wave function at fixed coupling constants.

A continuous-reduction model implying the dynamical suppression of linear superpositions of macroscopically distinguishable states, presented recently by Diòsi [Phys. Rev. A 40, 1165 (1989)], is investigated. The model exhibits appealing features; in particular, it relates reduction to gravity and contains no constants besides Newton’s gravitational constant G. It turns out, however, that the model is not fully consistent. A slight modification of this model is proposed, which overcomes the difficulties and retains partially its appealing features. The resulting model deals with systems containing distinguishable or identical constituents, allows the derivation from microdynamics of wave-packet reduction, and accounts for the emergence of definite macroscopic properties for macro-objects. Reduction is related to gravity in the same way as in Diòsi’s model, but a fundamental length must be introduced to avoid inconsistencies.

The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions ${\mathit{H}}_0$(α) indexed by the random variable α taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions.

The constrained optimization problem is reduced to an unconstrained optimization problem by the introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.

Precision lifetime measurements using laser excitation of a fast ion beam preexcited in a carbon foil are reported for two levels in N ii. The cascade-free decays of the fluorescence intensities give lifetimes of 0.249±0.004 and 0.267±0.010 ns for the N ii 2p3d ${}_{}{}^1{}_{}{}^{}$F° and 2p3s ${}_{}{}^1{}_{}{}^{}$P° levels, respectively. The lifetime result for the 2p3d ${}_{}{}^1{}_{}{}^{}$F° level—which is weakly repopulated by long-lived cascades—is in good agreement with beam-foil values and with the theoretical lifetime of McEachran and Cohen [J. Quant. Spectrosc. Radiat. Transfer 27, 119 (1982)]. The lifetime result for the 2p3s ${}_{}{}^1{}_{}{}^{}$P ° level—which is strongly repopulated by cascades—differs significantly from most of the previous experimental values but is in good agreement with the theoretical lifetimes of Luken and Sinanoglu [J. Chem. Phys. 64, 3141 (1976)], Beck and Nicolaides [Phys. Lett. 56A, 265 (1976)], McEachran and Cohen, and Fawcett [At. Data Nucl. Data Tables 37, 411 (1987)]. The f-value trend for the 2${\mathit{p}}^2$ ${}_{}{}^1{}_{}{}^{}$D–2p3s ${}_{}{}^1{}_{}{}^{}$P° transition along the C i sequence is discussed.

Energies of the $4s_{\frac{1}{2}}$, $4p_{\frac{1}{2}}$ and $4p_{\frac{3}{2}}$ states of Cu-like ions with nuclear charges in the range $Z=29-92\mathrm{}$ are calculated using relativistic many-body perturbation theory. These calculations include the lowest-order Dirac-Fock energies, second- and third-order Coulomb correlation corrections, the lowest-order retarded Breit interaction, second- and third-order correlation corrections to the Breit interaction, finite nuclear size corrections, and corrections for reduced mass and mass polarization. The order of magnitude of the omitted fourth- and higher-order correlation corrections is estimated by chaining second-order Brueckner orbitals. Using this estimate, we find that omitted correlation corrections to the ionization energies are less than the numerical error in the terms included in the calculation for $Z\ge 50$, and that omitted correlation contributions to the $4p_{\frac{3}{2}}-4s_{\frac{1}{2}}$ energy intervals are less than the numerical errors for $Z\ge 35$. The theoretical $4p_{\frac{3}{2}}-4s_{\frac{1}{2}}$ energy intervals, and the $4p_{\frac{3}{2}}-4p_{\frac{1}{2}}$ fine-structure intervals are compared with experiment to determine the QED contributions to the energies. The QED corrections inferred in this way are accounted for approximately by semiempirical values of the $n=4\mathrm{}$ Lamb shift.

The Schrödinger-Infeld-Hull factorization method is extended within the perturbation scheme in order to treat nonfactorizable Sturm-Liouville eigenequations in the same way as factorizable ones. It is shown that, provided suitable choices of the expansion basis set for the perturbing potential and for the associated perturbed ladder function are made, the solution of the factorizability condition associated with the perturbed eigenequation can be achieved by using an elementary finite difference calculus. An algebraic manufacturing process allowing the determination of the perturbed ladder and factorization functions, capable of handling any order of the perturbation and any type of factorization (Infeld-Hull types A to E), is given. This procedure, well adapted for computer algebra, allows an analytical determination of the perturbed eigenvalues and eigenfunctions without calculation of either the excited unperturbed eigenfunctions or any matrix element. This extension of the exact factorization method within the perturbation scheme can be applied to many model equations of current interest in quantum physics. Special attention is paid to perturbed factorizations that correspond to unperturbed ladder operators that are linear functions of the quantum number (types A to D). Illustrative applications are given. Particularly, the perturbed harmonic-oscillator ladder operators and eigenenergies are obtained in closed form.

An accurate calculation for ${}_{}{}^{1,3}{}_{}{}^{}$P autodetaching resonances in ${\mathrm{Ps}}^{\mathrm{-}}$ has been carried out using the complex-rotation method, which has the advantage of giving resonance position and width at the same time. The wave function is of the Hylleraas type with an available number of terms up to 1140. We find one ${}_{}{}^1{}_{}{}^{}$P and two ${}_{}{}^3{}_{}{}^{}$P resonances below the n=2 threshold and two ${}_{}{}^1{}_{}{}^{}$P and two ${}_{}{}^3{}_{}{}^{}$P resonances below the n=3 threshold of the positronium atom. Resonance parameters are compared with those obtained from scattering and adiabatic calculations where available.

Expectation values of ${\mathit{r}}^{\mathit{P}}$ for an arbitrary nl hydrogenic state are expressed in terms of state-independent coefficients that are determined by a simple recursion relation. Explicit results are given for negative P down to -16 and, by an algebraic transformation, for positive P up to 13. Closed-form expressions are obtained for the coefficients of the two highest-order terms in l multiplying each power of n. The result is an asymptotic expression valid for each P and sufficiently large l.

An analysis of various approaches to evaluating overlap and Coulomb matrix elements involving Gaussian-type-orbital primitive functions whose angular parts are solid spherical harmonics is given. Direct (angular momentum) coupled, direct uncoupled, and recursive evaluations are considered. The direct uncoupled evaluation involves some remarkably simple and transparent expressions. Using n-j symbols on a computer requires enough intermediate summations that uncoupled methods are best, particularly for higher angular momentum. Recursive methods may be competitive for low angular momenta.

Atoms at finite temperature and with a nucleus of finite extension are analyzed by a modification of the Thomas-Fermi model for T≠0. Applications to strange-matter atoms are included.

Relativistic Dirac-Fock and diagrammatic many-body perturbation-theory calculations have been performed on He, several He-like ions, Ne, and Ar. The no-pair Dirac-Coulomb Hamiltonian is taken as the starting point. A solution of the Dirac-Fock equations is obtained by analytic expansion in basis sets of Gaussian-type functions. Many-body perturbation improvements of Coulomb correlation are done to third order.

A simple method is described to determine ion quadrupole moments from spectroscopic data. Quadrupole moments of the ${\mathit{np}}^5$ ${}_{}{}^2{}_{}{}^{}$P${\mathrm{°}}_{3/2}$ ground states of 12 halogenlike ions are determined. The ions include five fluorinelike, three chlorinelike, two brominelike, and two iodinelike ions. The values of 〈${\mathit{r}}^2$〉 for the np electrons are determined from experimental energy separations within high angular momentum Rydberg configurations in the corresponding rare-gas-like ions. jl coupling is used in conjunction with an expression by Racah [Phys. Rev. 61, 537 (1942)] for the angular part of the quadrupole interaction between the ion core and the Rydberg electron. Hartree-Fock results for 〈${\mathit{r}}^2$〉 compare favorably with our values obtained from experimental data, although small systematic differences do exist, which increase with increasing ion mass. Limitations of the technique are also discussed. Such methods constitute a useful supplement to single-channel quantum-defect parametrization of energy levels for Rydberg states of atoms having cores with angular momentum ${\mathit{j}}_{\mathit{c}}$>1/2.

General, closed-form expressions are presented for the matrix elements of ${\mathit{ssS}}_{\mathit{K}}$(${\mathbf{s}}_{\mathit{i}}$,${\mathbf{s}}_{\mathit{j}}$) (rank K=0,1,2), tensors in the single-particle spin operators ${\mathbf{s}}_{\mathit{i}}$, in the unitary-group approach. Factorization of the matrix elements in terms of 6-j and 9-j coefficients is achieved through methods of spin-graphical analysis. Results are also obtained for the rank-2 spin-spin operator.

The nuclear quadrupole moments of ${}_{}{}^{33}{}_{}{}^{}\mathrm{S}$ and ${}_{}{}^{35}{}_{}{}^{}\mathrm{S}$ are studied using numerical multiconfigurational Hartree-Fock (MCHF) calculations on ${\mathrm{S}}^{\mathrm{-}}$${(}^2$${\mathit{P}}_{3/2}$). A newly developed finite-element MCHF program allowing very large configuration-interaction expansions is used. The 3${\mathit{p}}^5$ valence correlation decreases the electric field gradient with about 12%, but core-valence polarization increases the electric field gradient with about 14% of the Hartree-Fock value. The final values for the nuclear quadrupole moments of ${}_{}{}^{33}{}_{}{}^{}\mathrm{S}$ and ${}_{}{}^{35}{}_{}{}^{}\mathrm{S}$ become -0.0678(13) and 0.0471(9) b, respectively.

This paper reports an experimental investigation of muon-catalyzed fusion in pure deuterium by detection of dd fusion neutrons. Target temperatures of 25.5–150 K and gas densities of 2% and 5% liquid-hydrogen density were used. The rates λ̃F for ddμ formation from both hyperfine states of dμ atoms as well as the hyperfine transition rates λ̃FF′ were separated in a kinetic analysis of the observed time spectra. This measurement of the temperature dependence of all important rates in the ddμ catalysis cycle allows a comprehensive and quantitative test of the present theory of resonant molecule formation. In particular, the temperature behavior of ddμ formation from the dμ quadruplet state determines the binding energy ɛ11=-1966.1(3) meV of the participating ddμ state with unique accuracy. In general, convincing agreement between experiment and theory concerning the ddμ formation process is found, whereas the theoretical hyperfine rates, consisting of nonresonant and resonant contributions, exceed the experimental values by ∼40%.

We have obtained an exact analytical solution for a system of two particles moving under the action of a central harmonic-oscillator potential and the mutual Coulomb interaction augmented by a potential linear in the interparticle coordinate. The expressions for the single-particle density, the exchange-correlation energy quantities, and the correlation functions are thereby derived. Numerical results on the exchange-correlation contributions are presented using plots of various local energy densities as well as the corresponding global energy quantities for varying strengths of the oscillator potential.

In the present paper we report on measurements of the rate coefficient for radiative recombination between free electrons and bare ions. The measurements were done with merging beams of electrons and 2-MeV/amu fully stripped He, C, and F ions. Relative energies between 0 and 1 eV were considered. In the low-energy limit, calculations of the n and l distribution of the cross section, according to the ‘‘exact’’ theory by Stobbe, are presented for values of n up to 60. It is shown that corrections to the Bethe and Salpeter theory are needed for capture to the low-n states. The experimental data are compared with a calculation based on the Bethe and Salpeter theory as well as a calculation based on Stobbe’s theory for n≤3 and the Bethe and Salpeter theory for n>3.

A fully numerical multiconfiguration Hartree-Fock program has been modified for performance of calculations on atomic quasibound states using the complex-coordinate (dilation) technique. We present single-configuration calculations for the lowest ${}_{}{}^2{}_{}{}^{}$P° states of ${\mathrm{Be}}^{\mathrm{-}}$, ${\mathrm{Mg}}^{\mathrm{-}}$, ${\mathrm{Ca}}^{\mathrm{-}}$, ${\mathrm{Sr}}^{\mathrm{-}}$, and ${\mathrm{Ba}}^{\mathrm{-}}$, along with results of some small-scale multiconfiguration investigations of the Mg resonance. An instability originating in difficulty satisfying orthogonality constraints prevents a full multiconfiguration treatment of this resonance. Single-configuration dilated Hartree-Fock calculations predict ${}_{}{}^2{}_{}{}^{}$P° shape resonances for all these systems, though recent extensive calculations show the Ca, Sr, and Ba ions to be bound. Nonetheless, we think it is significant that the dilated numerical method can easily carry out fully variational computations on these heavy atoms, and our results, in conjunction with those of other studies, shed light on the importance of electron correlation in its effect upon the resonance parameters and its variation with progressively heavier alkaline-earth atoms. We also establish the feasibility of accounting for important correlation effects through the use of multiconfiguration approximations, though we have not attempted an accurate calculation in this work. Our Be, Mg, and Ca results agree well with previously published single-configuration dilated basis-set calculations, but no such calculations have been carried out for Sr or Ba. Finally, we investigate the dilational stability of the numerical method since this has been an important consideration in basis-set calculations. A simple exercise based on Cauchy’s theorem shows that analyticity of the Hartree-Fock radial wave functions implies that the energy and other physical quantities are independent of the dilation parameter, and a high degree of such independence is demonstrated in our numerical results.