This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.

Quasiexactly solvable problems are treated via the path-integral approach. The generalized Duru-Kleinert formalism involving a combination of a point-canonical transformation and a new-time transformation is applied to power-law and Morse-like quasiexactly solvable problems, and it is shown that these can be grouped into two families. A truncated kernel hypothesis is advanced, whereby the exactly solvable parts of the kernels of each problem within a given family can be transformed into one another. As an application, the ground-state energy eigenvalue and wave function of the N=0 generalized harmonic oscillator are derived from those of the N=0 generalized Coulomb problem.

A nonstandard realization of the su(1,1) algebra is used to extract a two-parameter class of scattering potentials as well as to calculate the reflection coefficient of the associated one-dimensional scattering problem in the spirit of the algebraic scattering theory. The nontrivial geometric content of such realizations is discussed, and an interesting connection with geometric phases is pointed out. It is argued that using larger noncompact groups, realizations related to non-Abelian geometric phases may be useful for obtaining analytical expressions for interaction terms corresponding to higher-dimensional scattering problems.

This paper addresses the nature and multiplicity of an optimally designed electric field for controlling quantum-dynamical processes. A rather general cost functional is considered, with only mild conditions called for amongst the various operators involved. An explicit upper bound on the magnitude of the controlling electric field is attained in terms of the norms of various operators entering into the control cost functional. An earlier work employing first-order perturbation theory arguments showed that, under rather mild assumptions, a denumerably infinite number of control-field solutions exists for the optimal control problem. In the present work, it is shown that through a bound on the remainder of the nonlinear terms in the expansion, this same conclusion concerning the control-field multiplicity continues to hold.

A variational method that uses the frequency and the energy shift as variational parameters is presented. The quantum-mechanical partition function is approximated by a formally simple expression, for a generalized anharmonic oscillator in one and many dimensions. The numerical calculations for a single quartic and two coupled quartic oscillators lead to nearly exact values for the free energy, the ground state, and the difference between the ground state and the first excited state.

Restrictive conditions on the class of allowed physical theories are drawn from the assumption that the predictions of quantum theory are valid for an experiment of the kind proposed by Greenberger, Horne, and Zeilinger [in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, 1989)]. It is shown that no theory can be compatible with the following four conditions. (1) The choices to be made by the three experimenters can be treated, in this context, as three independent free variables. (2) For each of the six possible local measurements under consideration, if that local measurement were to be performed, then exactly one of the alternative possible outcomes of this measurement must be selected as the actual outcome. (3) For each triad of measurements in a certain set of possible triads, if that triad were to be performed, then the corresponding triad of selected outcomes must satisfy the correlation condition predicted by quantum theory. (4) For each of the six possible local measurements, if that local measurement were to be performed, then the selected outcome must, according to the theory, be independent of which two experiments will later, in some frame of reference, be performed in the other two regions.

Using relativistic Green’s-function techniques we examined single-electron excitations from the occupied Dirac sea in the presence of strong external fields. The energies of these excited states are determined taking into account the electron-electron interaction. We also evaluate relativistic transition strengths incorporating retardation, which represents a direct measure of correlation effects. The shifts in excitation energies are computed to be lower than 0.5%, while the correlated transition strengths never deviate by more than 10% from their bare values. A major conclusion is that we found no evidence for collectivity in the electron-positron field around heavy and superheavy nuclei.

The relativistic hypervirial and Hellmann-Feynman theorems for the Klein-Gordon (KG) equation are used to construct Rayleigh-Schrödinger (RS) perturbation expansions to arbitrary order. The method is applied to the KG equation for a particle in an attractive Coulomb-type vector potential with perturbing vector or scalar potentials of the form λ${\mathit{r}}^{\mathit{k}}$, k=1,2, . . . . In the scalar case, such potentials are confining and the RS expansions exhibit Stieltjes behavior for k≥1 and Padé summability for k=1,2.

A source of recoil corrections to muonium hyperfine splitting which are linear in the logarithm of the electron-muon mass ratio is detected. The contribution is induced by the simultaneous insertion of muon and electron polarization loops in the external photon lines. This contribution is calculated analytically and is given by the numerical result δE=(${\mathrm{\alpha }}^2$Zα/π)${\mathit{E}}_{\mathit{F}}${0.6455[m/M]ln[M/m] +0.9212[m/M]}=0.011 37 kHz.

We derive simple expressions for the energy corrections to the Born-Oppenheimer approximation valid for a harmonic oscillator. We apply these corrections to the electronic and rotational ground state of ${\mathrm{H}}_2^{+}$ and show that the diabatic energy corrections are linearly dependent on the vibrational quantum numbers as seen in recent variational calculations [D. A. Kohl and E. J. Shipsey, J. Chem. Phys. 84, 2707 (1986)].

Experimental and theoretical studies of lifetimes and transition probabilities for doubly ionized mercury, Hg iii, are reported. Such data are currently of great astrophysical interest because observations with the Goddard high resolution spectrograph on board the Hubble space telescope have shown some 5${\mathit{d}}^9$6s-5${\mathit{d}}^9$6p transitions of Hg iii in the spectrum of the chemically peculiar star χ Lupi. We now report experimental lifetimes for some selected 5${\mathit{d}}^9$6p and 5${\mathit{d}}^8$6s6p levels, determined by beam-foil spectroscopy, and compare the results with theoretical transition probabilities based on relativistic multiconfiguration Dirac-Fock calculations.

The lifetime of the metastable autodetaching 1s2s2p ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{5/2}$ level of the ${}_{}{}^4{}_{}{}^{}\mathrm{He}_{}^{\mathrm{-}}{}_{}{}^{}$ ion has been measured by a time-of-flight technique to be 350±15 μs using a heavy-ion storage ring. Blackbody-induced photodetachment of this weakly bound level is shown to influence the decay rate by nearly 20%, an effect not discussed previously. The magnetic-field-induced mixing effects provide additional information about the lifetime of the close-lying ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{3/2}$ level, yielding τ${(}^4$${\mathit{P}}_{3/2}$)=12±2 μs. The observed lifetimes are consistent with previous experimental values, but the accuracy of the ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{5/2}$ lifetime is improved by a factor of 6, allowing a detailed test of theoretical predictions to be performed. The ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{5/2}$ lifetime is significantly shorter than most recent advanced theoretical results, but in agreement with a multiconfiguration Hartree-Fock approach accurately describing the initial bound ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{5/2}$ and the final continuum ${}_{}{}^2{}_{}{}^{}$${\mathit{F}}_{5/2}$ states.

For the Dirac equation in one space dimension with a potential of the Lorentz scalar type, we present a complete solution for the problem of constructing a transparent potential. This is a relativistic extension of the Kay-Moses method which was developed for the nonrelativistic Schrödinger equation. There is an infinite family of transparent potentials. The potentials are all related to solutions of a class of coupled, nonlinear Dirac equations. In addition, it is argued that an admixture of a Lorentz vector component in the potential impairs perfect transparency.

We report a measurement of the lifetime of the 2 ${}_{}{}^2{}_{}{}^{}$${\mathit{S}}_{1/2}$ state in one-electron krypton and a direct determination of the branching ratio for the M1 decay of this state. The branching-ratio measurement provides a sensitive test of the relativistic corrections to the decay rate. The lifetime measurement yields (36.8±1.4) ps and the branching ratio is found to be 0.356(15), both in good agreement with the theoretical values of 37.008 ps and 0.3643, respectively. From these measurements we also deduce the corresponding magnetic-dipole decay rate ${\mathit{A}}_{\mathit{M}1}$=9.68(55)×${10}^9$${\mathrm{s}}^{\mathrm{-}1}$ and the two-photon decay rate ${\mathit{A}}_{2\mathit{E}1}$ =1.750(78)×${10}^{10}$ ${\mathrm{s}}^{\mathrm{-}1}$.

A direct (nonvariational) solution of the Schrödinger equation for the ground state of the positronium negative ion is obtained with the correlation-function hyperspherical-harmonic (CFHH) method. Given the proper correlation function chosen from physical considerations, the CFHH method generates wave functions accurate in the whole range of interparticle distances that lead, in turn, to precise estimates of the expectation values of the Hamiltonian and of different functions of interparticle distances. The correlation function used was chosen to have proper electron-positron and electron-electron cusps as well as asymptotic behavior. The inclusion of 225 hyperspherical functions yields the nonextrapolated ground-state energy value of 0.262 005 058 atomic units, which is lower than the nonextrapolated energy values 0.262 004 895 and 0.262 005 056 calculated in works of Ho [J. Phys. B 16, 1503 (1983)] and Bhatia and Drachman [Phys. Rev. A 28, 2523 (1983)] but higher than the best variational value 0.262 005 069 obtained by Petenlenz and Smith [Phys. Rev. A 36, 5125 (1987)]. The accuracy of our value of 2.086 10 ±0.000 06 ${\mathrm{nsec}}^{\mathrm{-}1}$ for the two-photon annihilation rate is higher by an order of magnitude than obtained in the previous literature.

Coordinate scaling within the coupling-constant integration formula is used to determine simple approximations to the exchange-correlation energy functional ${\mathit{E}}_{\mathrm{xc}}$[ρ]. Correlation contributions to the kinetic energy, ${\mathit{T}}_{\mathrm{c}}$, are also determined by coordinate scaling. Calculations show that the approximations generated for ${\mathit{E}}_{\mathrm{xc}}$[ρ] give surprisingly good results for the species tested, considering the simplicity of the functionals, considering the fact that their forms are unorthodox, and considering the fact that they satisfy only a small fraction of the conditions known to be satisfied by the exact ${\mathit{E}}_{\mathrm{xc}}$[ρ]. The approximate functionals for ${\mathit{T}}_{\mathrm{c}}$ give reasonable values.

We present here a simple method for the direct evaluation of traces of p-order replacement operators (p-RO’s) calculated in a finite-dimensional, antisymmetric, and spin-adapted N-electron space. The procedure is based on the reduction of the operator order together with a performance of the summations through Slater determinants. Very useful and general formulas, which can be applied to RO’s of any order p≤N, are then derived.

In this paper, we used the multiconfiguration Dirac-Fock method to compute with high precision the 1${\mathit{s}}^2$2s2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_1$–1${\mathit{s}}^2$2s2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_0$ separation energy in berylliumlike ions, including the relativistic contribution to electron-electron correlations and radiative corrections. The effect of the hyperfine interaction on both the energy and lifetimes of those levels has been evaluated. In the absence of nuclear magnetic moment the ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_0$ level can decay to the ground state only by an E1-M1 transition, with a very low probability. We show that the hyperfine interaction increases dramatically the ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_0$ transition probability.

The curvature of the spherically averaged ground-state electron density of atoms is the subject of this study. We show that strict pseudoconvexity is the general structural property of the density of atoms which embodies all the structural features that have been empirically attributed to ρ(r). Rigorous upper and lower bounds for the second derivative of ρ(r) at the nucleus and at large distances are obtained. Within the bare Coulomb-field model atom, it is shown that the total electron density of the lowest set of closed shells is convex. Rigorous cusp conditions for the second derivative of the model atom density are derived and shown to be obeyed by model atoms with closed shells. Upper bounds that constrain the curvature of ρ(r) at the nucleus of open-shell model atoms are also developed. The Kohn-Sham formalism is used to obtain a general expression for the second derivative of ρ(r) at the nucleus.

The theory of a method to measure electric-dipole polarizabilities is presented. The method uses two-stage laser ablation to produce a pulsed beam of atoms from a solid target. A pair of slits makes the velocity distribution of the atomic beam nonuniform in a way that is well characterized. The light-force technique uses the polarization forces experienced by an atom in the intense, inhomogeneous electromagnetic field of a standing-wave laser to change the velocity distribution of the atomic beam. The large forces cause measurable Doppler shifts in the resonant frequency of the atoms. These frequency shifts change the amount of absorption of resonant light, yielding information about the change in the velocity distribution of the atoms. The detailed shape of the final absorption distribution is polarizability dependent. In the classical picture of the light force, the standing-wave electric field induces a time-varying dipole moment in each atom. Each atom then experiences a Lorentz force due to coherent interaction of the oscillating dipole moment with the time-varying magnetic field. The quantum-mechanical picture corresponds to the Kapitza-Dirac effect for atoms: an atom absorbs a photon from one of the two beams which form the standing wave, and is then stimulated to emit this photon by the other, counterpropagating beam. This paper provides a classical treatment of the light force experienced by an atom in a standing-wave light field; the polarizability is treated quantum mechanically. The theory presented here can be applied to atoms with a scalar polarizability, such as rubidium, and to atoms with significant tensor components, such as uranium. Experimental results from application of the light-force technique to a measurement of the polarizability of rubidium are also presented.