The Wigner function $W(x,p)\mathrm{}$ is related to the Weyl function $\tilde{W}\mathrm{}(X,P)\mathrm{}$ through a two-dimensional Fourier transform. Several properties for these functions are proved which lead naturally to an extended phase space $x-p-X-P.\mathrm{}$ Uncertainty relations for $\delta x\delta P$ and for $\delta p\delta X$ are studied. They reduce to the usual uncertainty relation in the special case of pure states, but are different for mixed states. Thermal density matrices obey these uncertainty relations as equalities.

The equations of motion for phase-space moments and correlations are derived systematically for quantum and classical dynamics, and are solved numerically for chaotic and regular motions of the Hénon-Heiles model. For very narrow probability distributions, Ehrenfest’s theorem implies that the centroid of the quantum state will approximately follow a classical trajectory. But the error in Ehrenfest’s theorem does not scale with $ħ,$ and is found to be governed essentially by classical quantities. The difference between the centroids of the quantum and classical probability distributions, and the difference between the variances of those distributions, scale as ${ħ}^2,$ and so are the true measures of quantum effects. For chaotic motions, these differences between quantum and classical motions grow exponentially, with a larger exponent than does the variance of the distributions. For regular motions, the variance of the distributions grows as $t^2,$ whereas the differences between the quantum and classical motions grow as $t^3.$

It is well known that environment may decohere a quantum bit (qubit) system immersed in it, making a quantum computation invalid. But the quantitative features of the decoherence seem to depend on both the constitution of the environment and the details of its coupling with the qubit system. In this paper, based on the dynamic approach for quantum measurement developed from the Hepp-Coleman model [K. Hepp, Helv. Phys. Acta 45, 237 (1972)], we generally model the environment as a collection of a large number of subsystems and then consider to what extent and in which way the environment and its coupling with the qubit system may affect a quantum computation process. In the weak-coupling limit, we find that as far as decoherence time is concerned, there is no essential difference between an environment of two-level subsystems and an environment of harmonic oscillators. This implies that there exists some universality independent of specific constitutions of environments. However, it is also shown that this is not true at finite temperature or in the case of strong coupling. So only if the coupling is weak and the temperature low does there exist the possibility of developing a universal scheme of controlling a qubit system such that the decoherence is avoided. The possible effect of environment on the efficiency of a quantum algorithm is also explicitly illustrated through the example of Shor’s prime factorization algorithm.

We present a class of fast quantum algorithms, based on Bernstein and Vazirani’s parity problem, that retrieves the entire contents of a quantum database $Y$ in a single query. The class includes binary search problems and coin-weighing problems. We compare the efficiency of these quantum algorithms with the classical algorithms that are bounded by the classical information-theoretic bound. We show the connection between classical algorithms based on several compression codes and our quantum-mechanical method.

We construct a unique optimal quantum device for turning a finite number of $d$-level quantum systems in the same unknown pure state σ into $M$ systems of the same kind, in an approximation of the $M$-fold tensor product of the state σ.

We investigate means to describe the nonlocal properties of quantum systems and to test if two quantum systems are locally equivalent. For this we consider quantum systems that consist of several subsystems, especially multiple quantum bits, i.e., systems consisting of subsystems of dimension 2. We compute invariant polynomials, i.e., polynomial functions of the entries of the density operator that are invariant under local unitary operations. As an example, we consider a system of two quantum bits. We compute the Molien series for the corresponding representation, which gives information about the number of linearly independent invariants. Furthermore, we present a set of polynomials that generate all invariants (at least) up to degree 23. Finally, the use of invariants to check whether two density operators are locally equivalent is demonstrated.

Very extensive multiconfiguration Dirac-Fock calculations including the transverse (Breit) interaction and quantum electrodynamics corrections have been carried out for all $3d$ transition metals to explain reliably the dependence of $K\beta$-to-$K\alpha$ x-ray intensity ratios on the changes in configurations of the valence electrons. For all considered atoms the greatest values of the $K\beta$-to-$K\alpha$ intensity ratios are for ${3d}^{m-2}{4s}^2$ configuration type, then for ${3d}^{m-1}{4s}^1,$ and the smallest values are for ${3d}^m$ type. It has been found that for each type of electronic configuration the $K\beta$-to-$K\alpha$ intensity ratios increase evidently with the atomic number and for a particular atom they are quite sensitive to the changes of the valence electronic configuration. The greatest relative increase of the $K\beta$-to-$K\alpha$ intensity ratios (4%) as a result of transition from electronic configuration of the ${3d}^{m-1}{4s}^1$ type to the ${3d}^{m-2}{4s}^2$ type takes place for Sc and the smallest relative increase (2%) takes place for Cu. The presented results make it possible to carry out reliable interpretation of various experimental $K\beta$-to-$K\alpha$ x-ray intensity ratios for $3d$ transition metals in their compounds and alloys and can also provide quantitative information about the changes of the valence electronic configurations of these metals in considered systems.

The hydrogen molecule $\left({\mathrm{H}}_2\right)$ contains the basic ingredients for understanding the chemical bond, even more so than the hydrogen molecule ion. ${\mathrm{H}}_2$ is studied in the context of nonlinear dynamics. The classical mechanics of ${\mathrm{H}}_2$ is studied in three dimensions with nine, six, and three degrees of freedom and in one dimension (two degrees of freedom). The semiclassical quantization is made using the Bohr-Sommerfeld rules and the Gutzwiller formula to calculate the eigenvalues of the doubly occupied symmetric excited states of ${\mathrm{H}}_2.$ An ab initio quantum calculation is performed and compared with semiclassical results. The difficulties that appear in those calculations are discussed, and a proposal of the experimental measure is made.

A highly stripped barium plasma is produced by 120-fs laser pulses irradiating a ${\mathrm{BaF}}_2$ target. The spectrum emitted by the plasma in the 9.10–9.36-Å wavelength range is recorded using a high-resolution spherically bent mica crystal. On the basis of the HULLAC atomic code, a level-by-level collisional-radiative model including autoionization and dielectronic capture processes is developed to calculate the wavelengths and intensities of the spectral lines emitted by each of the Cu-, Zn-, and Ga-like barium ions. $3d-nf$ $\mathrm{}(n=6,7\mathrm{})\mathrm{}$ spectral lines with different spectator electrons, previously observed only as unresolved transition arrays, are resolved. The theoretical results agree reasonably well with experiment. Best agreement is obtained for electron density and temperature of $5\times {10}^{21}{\mathrm{cm}}^{\mathrm{-}3}$ and 120 eV, respectively. The intensity ratios of the resolved Cu-like barium lines are shown to be useful tools for electron density and temperature diagnosis. This diagnostic method was not possible with low-resolution spectroscopy. It is found that at relatively low temperature and high density as in the present experiment, the relative intensities of lines within each ionization state are independent of the ion density ratio of adjacent ionization states.

An energy-dependent wave operator theory of quantum dynamics is derived for time-independent and time-dependent Hamiltonians. Relationships between Green’s functions, wave operators, and effective Hamiltonians are investigated. Analytical properties of these quantities are especially relevant for studying resonances. A derivation of the relationship between the Green’s functions and the ${(t,t}^{\prime })$ method of Peskin and Moiseyev [J. Chem. Phys. 99, 4590 (1993)] is presented. The observable quantities can be derived from the wave operators determined with the use of efficient iterative procedures. As in the theory of Bloch operators for bound states, the theory is based on a partition of the full Hilbert space into three subspaces: the model space, an intermediate space, and the outer space. On the basis of this partition an alternative definition of active spaces currently considered in large scale calculations is suggested. A numerical illustration is presented for several model systems and for the Stark effect in the hydrogen atom.

The carbon $1s$ photoelectron spectrum of gas-phase ethylene has been measured with high resolution in order to study changes in vibrational structure arising from core photoionization through a continuum resonance. By evaluating the changes in vibrational intensity distribution we are able to determine that the ionization is non-Franck-Condon for energies close to the maximum of the broad resonance in the ethylene absorption spectrum. From these measurements it is possible to investigate the specific effect on the vibrational intensity distribution arising from the resonance. We find that the intensity of the C-H stretch vibrational mode is constant as the photon energy is scanned through the resonance, while the intensity of vibrations associated with the C-C bond varies with photon energy. This shows the existence of a shape resonance which, according to the scattering model of the shape resonance, must be primarily associated with the C-C bond rather than the C-H bond.

Multiconfiguration Dirac-Fock (MCDF) wave functions for a specific J quantum number may not reduce to the appropriate nonrelativistic limit for the L and S quantum numbers. Transition probabilities calculated from such MCDF wave functions for spin-forbidden transitions are unreliable. Remedies for this problem are discussed, including the advantages of using nonorthogonal radial orbitals. It is also shown that transition probabilities for weak transitions are sensitive to the way the Breit interaction is introduced.

We have measured the hydrogen-deuterium isotope shift in the energy of the lowest ${}^1{D}^e$ autodetaching resonance in the two-photon absorption spectrum of ${\mathrm{H}}^{-}$. The ${}^1{D}^e$ resonance in negative deuterium lies $2.0\pm 0.5$ meV, with a possible systematic error of 0.4 meV, above the corresponding resonance in negative hydrogen when measured with respect to the ion ground state. From the isotope shift, we obtain a mass polarization of $2.4\pm 1.1$ meV, with a possible sytematic error of 0.8 meV, for the ${}^1{D}^e$ in ${\mathrm{H}}^{-}$ under the assumption that the Fano shape parameters, $q$, for the two isotopes are the same. Recent theoretical calculations give the mass polarization for this resonance in ${\mathrm{H}}^{-}$ as $-0.1$ meV. The generalized absolute cross section of two-photon absorption at the peak of the ${}^1{D}^e$ is found to be ${(3.2\mathrm{}}_{-1.2}^{+1.8\mathrm{}})\times {10}^{-49}\hspace{0.5em}{\mathrm{cm}}^4$ s. The measured asymmetry parameters ${\beta }_2$ and ${\beta }_4$ of the photodetachment process are $1.9\pm 1.2$ and $2.21\pm 0.45$, respectively, consistent with a pure $D$-wave distribution. Upper bounds on the second recursion of the lowest ${}^1{S}^e$ resonance and three-photon excess photon detachment are $2.8\times {10}^{-49}\hspace{0.5em}{\mathrm{cm}}^4\mathrm{s}$ and $3.1\times {10}^{-78}\hspace{0.5em}{\mathrm{cm}}^6{\mathrm{s}}^2,$ respectively.

Using the classical mechanical perturbation approach and semiclassical quantization rules we explain the spectrum of atomic hydrogen in a weak magnetic field in which fine structure of levels cannot be neglected. The general pattern of the spectrum turns out to be very different from the nonrelativistic scheme. In particular, we point out the presence of another type of “exponentially narrow” doublet.

The performance of commonly used correlation and exchange-correlation functionals has been studied for highly charged atomic ions. Calculated energies have been compared with exact nonrelativistic energies of atomic species with 2–18 electrons for nuclei from $Z=2\mathrm{}$ (He) to $Z=26\mathrm{}$ (Ni). It is found that all density functionals considered here reveal problems in the description of correlation effects as a function of nuclear charge Z.

The photoabsorption spectra of atoms in a static external electric field shows modulations from recurrences: electron waves that go out from and return to the vicinity of the atomic core. Closed-orbit theory predicts the amplitudes and phases of these modulations in terms of closed classical orbits. A classical scaling law relates the properties of a closed orbit at one energy and field strength to its properties at another energy and field strength at fixed scaled energy $\epsilon {=EF}^{-1/2}.$ The scaling law states that the recurrence strength of orbits along the electric field axis scale as $F^{1/4}.$ We show how this law fails near bifurcations when the effective Planck constant $\widehat{ħ}\equiv ħF^{1/4}$ increases with increasing field at fixed $\epsilon .$ The recurrences of orbits away from the axis scale as $F^{1/8}$ in accordance with the classical prediction. These deviations from the classical scaling law are important in interpreting the recurrence spectra of atoms in current experiments. This leads to an extension of the uniform approximation developed by Gao and Delos [Phys. Rev. A 56, 356 (1997)] to complex momenta.

The positronium molecule $\left({\mathrm{Ps}}_2\right)$ has not been experimentally observed yet because its tiny (4.5 eV) binding energy cannot be detected when the molecule annihilates by emitting two photons with energy of 0.51 MeV each. It is shown in this paper that the electric dipole transition between the recently found $L=1\mathrm{}$ excited state and the $L=0\mathrm{}$ ground state with its characteristic photon energy of 4.94 eV is a clear signature of the existence of the positronium molecule and the possibility of its experimental observation is realistic. The probability of this transition is about 17% of the total decay rate. Another Coulomb four-body system containing positron HPs (the positronium hydride or hydrogen positride), is also included for comparison.

The paper presents Λ splittings and q factors in the NaK $D{}^1\Pi$ state, directly measured from the electric radio-frequency-optical double resonance (RF-ODR) in laser-induced fluorescence (LIF) for a number of vibrational states $v=1\mathrm{}–22$ with definite rotational levels J between 7 and 46. Permanent electric dipole moment values (d) have been obtained by measuring in LIF spectra the relative intensities of “forbidden” lines caused by dc Stark effect induced $e/f$ mixing in the ${}^1\Pi$ state, with their subsequent processing, which allowed us to obtain the $q/d$ ratio. A possible influence of the hyperfine structure on the RF-ODR signal and relative intensities has been calculated, showing that this influence can be neglected. The $q\left(v\right)\mathrm{}$ values exhibited a decrease from $q\left(1\right)=1.529\mathrm{}\times {10}^{-5}{\mathrm{cm}}^{\mathrm{-}1}$ to $q\left(22\right)=1.171\mathrm{}\times {10}^{-5}{\mathrm{cm}}^{\mathrm{-}1},$ which has been explained by an increase of the difference potential between $D{}^1\Pi$ and $C{}^1{\Sigma }^{+}$ states with internuclear distance (R); the respective L-uncoupling matrix element was evaluated as 1.87. It was shown, both by semiempirical treatment and population analysis of ab initio molecular wave functions, that considerable $\pi d$ and $\sigma d$ configuration admixtures are present in the $D{}^1\Pi$ and the $C{}^1{\Sigma }^{+}$ states. For the $B{}^1\Pi$ state, it was demonstrated that Λ doubling is caused by two competing perturbers $(A{}^1{\Sigma }^{+}$ and $C{}^1{\Sigma }^{+}),$ yielding q factors of $\sim -2\times {10}^{-6}{\mathrm{cm}}^{\mathrm{-}1},$ in agreement with high-resolution spectroscopy data given in the literature; single-configuration approximation is valid for interacting $B{}^1\Pi (\sigma {3s}_{\mathrm{Na}},\pi {4p}_{\mathrm{K}})\sim A{}^1{\Sigma }^{+}(\sigma {3s}_{\mathrm{Na}},\sigma {4p}_{\mathrm{K}})$ states. The measured $d\left(v\right)\mathrm{}$ values, which varied from 6.6 to 4.6 D, have been used to obtain the empirical $D{}^1\Pi$ state $d\left(R\right)\mathrm{}$ function for $R=6\mathrm{}–12\mathrm{a}.\mathrm{u}.\mathrm{}$ by means of an improved instability-free inversion procedure exploiting a special functional form. Two independent series of ab initio all-electron calculations of $d\left(R\right)\mathrm{}$ and $d\left(v\right)\mathrm{}$ have been performed for the $D{}^1\Pi$ and $B{}^1\Pi$ states of NaK. First, d values were computed as expectation values of the electric dipole operator with conventional multireference configuration-interaction wave functions. Second, the finite-field (FF) technique, combined with a multipartitioning perturbation theory (MPPT) treatment of electronic eigenstates, was applied for the calculation of $d\left(R\right)\mathrm{}$ functions. The FF-MPPT calculations showed excellent agreement with experimental $D{}^1\Pi d\left(v\right)\mathrm{}$ values obtained in the present work, as well as the proximity to experimental $B{}^1\Pi d\left(v\right)\mathrm{}$ values given in the literature, thus showing that, as distinct from the ground state, it is important to account correctly for effective interactions of valence electrons arising from core-valence correlations, which could not be done properly with previously used pseudopotential techniques. The experimental d and q values dropping out from a smooth $v$ dependence have been considered as perturbed by $D{}^1\Pi \sim d{}^3\Pi$ interaction and exploited to evaluate respective $d_t$ and $c_t$ values for the perturbing $d{}^3\Pi$ state.

We report the observation of electric-dipole-forbidden ${2p}_{1/2}{-2p}_{3/2}$ transitions in berylliumlike, boronlike, carbonlike, nitrogenlike, oxygenlike, and fluorinelike uranium. Our measurement identified nine magnetic dipole transitions, as well as one strong electric quadrupole transition in carbonlike ${\mathrm{U}}^{86+}$ that has not been observed in lower-Z ions because of collisional quenching. The measurement was carried out with a high-resolution crystal spectrometer, and an accuracy as good as 35 ppm was obtained in the determination of the transition energies. The transitions are significantly less affected by quantum electrodynamical effects than analogous electric-dipole-allowed transitions in these ions. The data thus provide benchmarks for theoretical approaches of electron-electron correlation effects in the high-Z limit $Z\alpha \approx 1$ that complement earlier measurements of $2s-2p$ transitions in such highly charged ions. The accuracy of the present measurements, however, was sufficient to determine the residual contributions from quantum electrodynamics (about 2 eV) with an accuracy of 5%, i.e., with an accuracy comparable to that of the best measurements of such contributions to the $1s$ ground level in hydrogenic ${\mathrm{U}}^{91+}.$ The contributions from quantum electrodynamics are in large part due to the vacuum polarization terms. The ${2p}_{1/2}{-2p}_{3/2}$ transition energies thus provide a handle for testing the accuracy of vacuum polarization terms nearly independent of the terms arising from the electron self energy. Results from collisional-radiative calculations are presented that show that the forbidden lines are almost exclusively produced by indirect processes, i.e., radiative cascades, radiative electron capture, and the ionization of ${2p}_{1/2}$ electrons. This is in stark contrast to the electric dipole-allowed transitions, which are mostly populated by direct electron-impact excitation.

The line-integral method developed by van Leeuwen and Baerends [Phys. Rev. A 51, 170 (1995)] is applied to the calculation of the differences of correlation energy functional values $\Delta E_c^{\mathrm{DFT}}{=E}_c^{\mathrm{DFT}}\left[{\rho }_{\mathrm{HF}}\right]-E_c^{\mathrm{DFT}}\left[{\rho }_{\mathrm{exact}}\right],$ where ${\rho }_{\mathrm{HF}}$ is the Hartree-Fock density and ${\rho }_{\mathrm{exact}}$ is the near-exact one (DFT is density-functional theory). From the Kohn-Sham wave functions yielding Hartree-Fock and the near-exact densities, the corresponding noninteracting kinetic energies and the exchange energies are calculated. An approximate relation between $E_c^{\mathrm{DFT}}\left[{\rho }_{\mathrm{HF}}\right]$ and the conventional quantum chemistry correlation energy is presented, accurate to $<~4\mu \hspace{0.5em}\mathrm{hartree}$ for the isoelectronic series of He, and Li, and for the Be atom.