The nonlocality inherent in quantum mechanics has been tested experimentally, but the nonlocal interpretation of all the relevant experiments can be challenged. All past tests of Bell’s inequalities have required supplementary assumptions, without which the experimental results satisfy the inequalities, in agreement with the notion of local realism. There are basically three loopholes that need to be closed; neither the cascade sources nor the parametric down-conversion sources of correlation photons used to date are capable of closing all of them, even if the detectors used had been 100% efficient. We propose a two-crystal down-conversion source, relying on type-II collinear phase matching, which should permit a violation of Bell’s inequalities without the need for supplementary assumptions. As the source can produce a true singletlike state, it is also relevant for quantum cryptographic applications.

A beam splitter is assumed to separate an incident beam of light with indeterminate phase into a reflected and a transmitted beam. In analogy with classical physics, a homodyne approach similar to that suggested by Noh, Fougeres, and Mandel [Phys. Rev. Lett. 67, 1426 (1991)] is used to measure the phase of each beam at two distant locations. The nonlocal collapse of the wave function due to such a phase measurement made on one of the two beams is shown to reduce the state of the remaining beam to a coherent state (to a good approximation) with the corresponding phase. However, these homodyne phase measurements also exhibit several nonclassical features, including an oscillatory fine structure, which complicate their interpretation as the phase of the field.

A rule is proposed to calculate the geometric phase accumulating in the course of a sequential measurement process. This rule is based on the assumptions of a sequential conditioning and a geodesic hypothesis. The dependence of this phase on the type of the applied measurements is explicated.

In this paper a method for eigenvalue problems in quantum mechanics is developed. Principles of the first and second kinds of balance in a varying-parameters method take advantage of the standard perturbation theory and the standard variational principle. According to this method, we not only obtain the best approach for obtaining eigenvalues and wave functions, but we also determine structures of a quantum system. The extended Hamiltonian in a varying-parameters method is different from a standard Hamiltonian. The parameters are inserted into a Hamiltonian by adding and subtracting appropriate terms which contain the essential parameters. Thus it becomes possible to study the inner structure of a quantum system by applying principles of balance. In order to interpret the physical meanings of balance parameters, several examples are described. Applications are also made to the helium atom, heliumlike ions, and the lithium atom. We theoretically predict energies and structure parameters and obtain good agreement with experimental data, especially for high-orbit electrons (clearly this is a character of perturbation theory). It is quite interesting that the theoretical predictions of energy levels of parahelium in the S state are lower than the energy levels of orthohelium and that theoretical predictions of singlet (triplet) states are close to the experimental data of triplet (singlet) states. It would seem that the experimental data of triplet and singlet are reversed. Is that possible?

A comprehensive study is presented on the Weyl-Wigner formalism for rotation-angle and angular-momentum variables: the elements of kinematics are extended, the elements of dynamics are established, and the implications of rotational perodicity and angular-momentum quantization are investigated. Particular attention is paid to discreteness, and two of its consequences are emphasized: the importance of evenness and oddness, and the need to use two difference operators in a discrete domain, whereas one differential operator suffices in a continuous domain. These consequences are shown to strongly distinguish the Weyl-Wigner formalism for rotation-angle and angular-momentum variables from the well-known Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. The point is made clear that the first of these formalisms cannot be regarded as a trivial and straightforward extension of the second. The rotational Wigner function is derived as the only bilinear form of the state vector that is real, has the natural invariances for rotational motion, and yields the correct distributions for the rotation-angle and angular-momentum variables as well as the appropriate expression for the transition probability between states. The conditions for its uniqueness are thus established. Its properties are described in detail and, in particular, its uniform boundedness is demonstrated. The rotational Wigner function and the associated correspondence between quantum operators and classical-like functions, as well as the relations they obey, are explored and are written so as to clearly exhibit the distinct contributions of evenness and oddness in the discrete domain of the angular-momentum eigenvalues, thus providing a most natural way to account for periodicity.

Using the derivative, which acts on the continuous rotation-angle variable, and the forward and backward differences, which act on the discrete angular-momentum variable, the equation of motion for the rotational Wigner function is established. This equation is detailed for the following Hamiltonian forms: those that depend only on the angular-momentum variable, including, in particular, the free rotator, and those that are in the cosine of the rotation-angle variable. It is verified that the Weyl-Wigner formalism for rotation-angle and angular-momentum variables has the correct nonperiodic limit and that it properly reduces to the Weyl-Wigner formalism for Cartesian-position and linear-momentum variables. In order to illustrate the formalism, a careful analysis is carried out for the rotational Wigner function representing the energy eigenstates of a hindered rotator whose Hamiltonian is the sum of a term in the absolute value of the angular-momentum variable with a term in the cosine of the rotation-angle variable. For this hindered rotator, and within the approximation of a large absolute value of the angular-momentum variable, the equation of motion for the rotational Wigner function is solved for its stationary solutions, and the time-independent Schrödinger equation is also solved.

The semiclassical quantum theory is investigated with conformal mapping to analyze the energy levels for a central force potential in two dimensions. The conformal mapping produces a modification in the effective potential which provides the correct phase for the WKB wave function. With this modification in the effective potential the semiclassical quantum theory, the so-called WKB approximation, yields the exact energy spectrum for a central force potential in two dimensions, such as the parabolic potential, the attractive Coulomb potential, and the donor impurity states under a plane magnetic field in a two-dimensional system. Among these, the result for the last one is compared with experimental results and other numerical results.

We demonstrate that although the well-known analogy between the time-independent solutions for two-dimensional tunneling (e.g., frustrated total internal reflection) and tunneling through a one-dimensional potential barrier cannot, in general, be extended to the time domain, there are certain limits in which the delay times for the two problems obey a simple relationship. In particular, when an effective mass is chosen such that ${\mathit{mc}}^2$=ħω, the ‘‘classical’’ traversal times for allowed transmission become identical for a photon of energy ħω traversing an air gap between regions of index n and for a particle of mass m traversing the analogous square barrier of height ${\mathit{V}}_0$ in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E≊${\mathit{V}}_0$ (θ≊${\mathrm{\theta }}_{\mathit{c}}$) and for E≫${\mathit{V}}_0$ (θ≪${\mathrm{\theta }}_{\mathit{c}}$). (For a smoothly varying potential or index of refraction, the agreement persists for all values of E where the WKB approximation applies.) The same relation serves to equate the quantum-mechanical ‘‘dwell’’ times for any values of E and ${\mathit{V}}_0$. On the other hand, in the ‘‘deep tunneling’’ limit, E≪${\mathit{V}}_0$ (θ≊π/2), one must choose ${\mathit{mc}}^2$=${\mathit{n}}^2$ħω in order to make the group delays equal for the two problems. These equivalences simplify certain calculations, and the two-dimensional analogy may also be useful for geometrically visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that the equality of the group delays for transmission and reflection for lossless barriers follows from a simple intuitive argument based on time-reversal invariance, and discuss the extension of the result to the case of lossy barriers.

A two- or three-dimensional quantal isotropic anharmonic oscillator is treated by means of the phase-integral method of Fröman and Fröman. The generalized Bohr-Sommerfeld quantization condition for the radial wave function is expressed in terms of complete elliptic integrals up to the fifth order of the phase-integral approximation. The quantization condition is solved numerically, and energy levels are obtained for various quantum numbers. Comprison with numerically exact results is also made.

A recurrence method for analytical and numerical evaluation of tunneling, transmission, and reflection amplitudes is developed. As the first step, a rule for composition of two arbitrary scatterers separated by a region of constant potential is obtained. Transmission and reflection amplitudes for this double-barrier potential are expressed in terms of transmission and reflection amplitudes for its subparts. As the length of the constant-potential region can be arbitrary and the subparts of a potential may, in turn, be arbitrary segmented potentials, one obtains formulas recurerence formulas which express the scattering amplitudes for the arbitrary segmented potential via the scattering amplitudes for the subparts into which the complete potential can be divided. The efficiency of the method is demonstrated by solving analytically the problem of scattering from locally periodic potentials. Since an arbitrary potential can be approximated by a set of infinitely narrow rectangular barriers, the recurrence formulas can be applied to any potential, giving, in the limit of zero-width segments, differential equations for transmission, and reflection amplitudes.

Three-photon excitation of autoionizing states between the ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{3/2}$ and ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{1/2}$ ionic limits of atomic argon, krypton, and xenon has been investigated using multiphoton ionization in combination with supersonic expansion techniques. As the result of the high signal-to-noise ratios and energy resolution achieved in these experiments, we have been able to study autoionizing states in argon with multiphoton excitation, and have considerably augmented the knowledge of the autoionizing states of krypton and xenon. All atoms show (in ${\mathit{j}}_{\mathit{c}}$l coupling notation) the odd ns’[1/2${]}_1$, nd’[3/2${]}_1$, and ng’[7/2${]}_3$ autoionizing series, while the nd’[5/2${]}_3$ series can only be observed in xenon. The analysis of the line profile of the autoionizing resonances has enabled a detailed comparison of the present results with those obtained with other experimental techniques and predictions by theoretical studies.

The lifetimes of the 6p ${}_{}{}^2{}_{}{}^{}$ ${\mathit{D}}_{5/2}^{\mathit{o}}$, 6p ${}_{}{}^4{}_{}{}^{}$${\mathit{P}}_{5/2}^{\mathit{o}}$, 6p ${}_{}{}^4{}_{}{}^{}$ ${\mathrm{D}}_{7/2}^{\mathit{o}}$, 6p ${}_{}{}^4{}_{}{}^{}$${\mathit{D}}_{5/2}^{\mathit{o}}$, and 6p ${}_{}{}^2{}_{}{}^{}$${\mathit{F}}_{7/2}^{\mathit{o}}$ levels in Xe ii have been measured with a beam-laser technique. A detailed spectroscopic investigation of the fluorescence channels of the excited states has been performed to confirm the identifications. As a consequence of some reassignments experimental lifetime values can now unambiguously be ascribed to these levels.

Relativistic and QED corrections are calculated by using a direct solution of the Schrödinger equation for the 2 ${}_{}{}^1{}_{}{}^{}$S excited state of the helium atom obtained with the correlation-function hyperspherical-harmonic method. Our extremely accurate nonvariational results for relativistic, QED, and finite-size corrections coincide exactly (up to 0.000 03 ${\mathrm{cm}}^{\mathrm{-}1}$) with the values obtained in precision variational calculations of Drake [Nucl. Instrum. Methods Phys. Res. B 5, 2207 (1988)] and Baker, Hill, and Morgan [in Relativistic, Quantum Electrodynamic and Weak Interaction Effects in Atoms, edited by Walter Johnson, Peter Mohr, and Joseph Sucher, AIP Conf. Proc. No. 189 (AIP, New York, 1989), p. 123] for both infinite and finite nuclear masses. This confirms that a discrepancy of 0.0033 ${\mathrm{cm}}^{\mathrm{-}1}$ between theory and experiment is not a result of an inaccuracy of variational wave functions, but is rooted in our inadequate knowledge of the QED operators. A better understanding of the different QED contributions to the operators (such as, for example, a more precise estimate of the Bethe logarithm) is therefore needed to explain the discrepancy.

Relativistic and finite-size corrections are calculated by using an accurate direct solution of the Schrödinger equation [the correlation function hyperspherical harmonic (CFHH) method] for the ground state of the helium atom. In the CFHH method the solution is a product of a correlation function and of a smooth factor expanded into hyperspherical harmonic functions. Given the proper correlation function, chosen from physical considerations, the method generates wave functions, accurate in the whole range of interparticle distances, which leads in turn to precise estimates of the relativistic operators. Our nonvariational results are compared with those obtained in recent variational calculations.

Isotope shifts and hyperfine structure in the 369.4-nm 6s-6${\mathit{p}}_{1/2}$ resonance line of the single-valence-electron system ${\mathrm{Yb}}^{+}$ have been determined with an accuracy of about 1 MHz by Doppler-free saturated absorption spectroscopy in a sputtered vapor. Ab initio many-body perturbation theory calculations in the coupled-cluster approach were then used to evaluate the electronic field shift factor, F=-14.9(2) GHz ${\mathrm{fm}}^{\mathrm{-}2}$, and to estimate the specific mass shift (SMS) factor, ${\mathit{K}}^{\mathrm{SMS}}$=(1±1)${\mathit{K}}^{\mathrm{NMS}}$, where NMS is the normal mass shift. The uncertainty in the calculated F factor is based on the level of agreement between the hyperfine structure constants calculated for 6s and 6${\mathit{p}}_{1/2}$ states using the same wave functions as for the F-factor calculation and the experimentally determined hyperfine-structure constants. The calculated F and ${\mathit{K}}^{\mathrm{SMS}}$ factors have been used to extract values for the difference in mean-square charge radius, δ〈${\mathit{r}}^2$${\mathrm{〉}}_1^{\mathit{A}}$,${\mathit{A}}_2$, between isotope pairs ${\mathit{A}}_1$,${\mathit{A}}_2$, and the related nuclear charge distribution parameter ${\lambda }_1^{\mathit{A}}$,${\mathit{A}}_2$, which are just within the uncertainties of the tabulated values of Aufmuth et al. [At. Data Nucl. Data Tables 37, 455 (1987)] based on semiempirical estimates of F and assumed values of ${\mathit{K}}^{\mathrm{SMS}}$.

We derive momentum-transfer dispersion relations by showing that at fixed impact energy the electron-atom differential cross sections are analytic functions of the momentum transfer squared ${\mathit{K}}^2$ in a complex plane cut from -∞ to 0, along the real axis. It is therefore natural to introduce sets of interpolating rational functions of ${\mathit{K}}^2$ to fit experimental data. The most suitable are the Padé approximations. We find that the zeros and the poles of these approximations split into two families. One family is made of poles and zeros that sit on the cut, yielding a good simulation of it. The other family takes care of the noise in the data: poles and zeros appearing in pairs very near each other. We can therefore first filter the noise by eliminating those pairs. Among the remaining poles, one pole is extremely near ${\mathit{K}}^2$=0 for the inelastic differential cross sections. We apply this technique to recompute both elastic and inelastic cross sections for Xe, Kr, and Ar atoms, at impact energies of 100, 400, and 500 eV. In this way, we get the optical oscillator strength for two optically connected states. Our results are compared with other experimental as well as theoretical results.

We discuss the magnetic toroidal dipole moments (anapole moments) of diatomic molecules by analyzing a calculable ‘‘toy model’’ for a heteronuclear diatomic molecular ion. The model consists of a single unpaired electron with spin σ, in the electrostatic potential of two different point charges; all spin-dependent interactions are neglected. We show that this model predicts a toroidal dipole moment of order α${\mathit{ea}}_0^2$ oriented in the direction n^×σ, where n^ is the molecular axis. A simple relation is derived between the three static dipole moments of this model: electric, magnetic, and toroidal. Numerical results are given for a particular choice of model parameters. In addition to the usual magnetic dipole field, there is a toroidal magnetic field inside the molecule, which we evaluate for this model. The two fields are similar in strength but have opposite behavior under spatial inversions, so that the magnetic field, as well as the electric field, fails to have a center of inversion.

Numerical and analytical procedures for the calculation of exchange integrals with Slater-type orbitals are reviewed and checked. Algorithms for improving the calculation of the auxiliary functions employed in both kinds of procedures are introduced and their efficient implementation is discussed. Finally, the numerical behavior of these procedures is analyzed paying special attention to their accuracy thresholds and computational cost.

The probe spectrum of a closed two-level Zeeman-degenerate atom interacting with a linearly polarized coherent laser field is studied. Results for the cycling transition of a cesium atom are presented and compared to the experimental data of Lounis et al. [Phys. Rev. Lett. 69, 3029 (1992)]. We further show that the resonance derived by Grynberg, Vallet, and Pinard [Phys. Rev. Lett. 65, 701 (1990)] is due to stimulated Raman scattering.

High-resolution inverse Raman experiments for ${\mathrm{H}}_2$ diluted in Ar have revealed unusual Q-line profile features at various concentrations which have been interpreted in terms of speed-changing collisions. A more general approach including both speed- and phase-changing collisions has led to an alternative interpretation of these features. In order to further analyze the role of these various collisional mechanisms, new experiments have been performed. Pure ${\mathrm{H}}_2$ and ${\mathrm{H}}_2$ perturbed by Ne, Ar, and Xe have been investigated at various temperatures between 295 and 800 K.

Many functions of physical interest are defined by means of recursion relations where a function of a given rank can be expressed in terms of the functions of lower rank. It is also usually possible to define approximation schemes which, with natural modifications, can be applied to evaluate any function of the hierarchy. Typical examples are density functions, propagators, and Green’s functions, and the approximation approaches to them are often based on perturbation theory. The attempt of this work is to clarify the connection between the approximation schemes for propagators and the hierarchy of linking equations. The concept of scheme consistency is introduced and used to extend available approximation schemes. As a specific example the theory is worked out for the particle-particle propagator, a quantity describing the simultaneous removal of two particles from a many-particle system. It is expected that the derived working equations are of importance in the description of double-ionization processes where each of the two removed electrons originates from a different part of the system.