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We investigate the prospects of combining a standard momentum space approach for ultracold three-body scattering with efficient coordinate space schemes to solve the underlying two-body problem. In many of those schemes, the two-body problem is numerically restricted up to a finite interparticle distance ${r}_{\mathrm{b}}$. We analyze the effects of this two-body restriction on the two- and three-body level using pairwise square-well potentials that allow for analytic two-body solutions and more realistic Lennard-Jones van der Waals potentials to model atomic interactions. We find that the two-body $t$ operator converges exponentially in ${r}_{\mathrm{b}}$ for the square-well interaction. Setting ${r}_{\mathrm{b}}$ to 2000 times the range of the interaction, the three-body recombination rate can be determined accurately up to a few percent when the magnitude of the scattering length is small compared to ${r}_{\mathrm{b}}$, while the position of the lowest Efimov features is accurate up to the percent level. In addition, we find that with the introduction of a momentum cutoff, it is possible to determine the three-body parameter in good approximation even for deep van der Waals potentials.

We consider higher-order finite-nuclear-size (FNS) contributions to the energy levels of light ordinary and muonic hydrogenlike atoms. Relativistic corrections to the leading FNS term have been known in terms of an integration in coordinate space. The related results are model dependent. In the meantime, if the data on the nuclear structure are obtained from electron-nucleus scattering, it is a representation of the electric charge form factor in momentum space rather than the charge density. An example is the case of a proton as a nucleus. Often the electric form factor is presented in terms of a dispersion integral or a Pad\'e approximation. We present explicit results for such representations for leading finite-nuclear-size higher-order (in $Z\ensuremath{\alpha}$ and in $Z\ensuremath{\alpha}m{R}_{N}$) corrections beyond the leading FNS term.

We present a theoretical consideration of the radiative two-photon finite-nuclear-size contributions to the Lamb shift in a light muonic atom. They are of the order of $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{5}m$ and may depend differently on $m{R}_{N}$. The contributions are due to the electronic and muonic vacuum polarization and the muon self-energy. The consideration is done within the external field approximation. We also found the leading logarithmic finite-nuclear-size contributions in the next order. One of them is of the order $\ensuremath{\alpha}{(Z\ensuremath{\alpha})}^{6}{ln}^{2}(Z\ensuremath{\alpha})\phantom{\rule{0.16em}{0ex}}{(m{R}_{N})}^{2}m$ and the other is of the order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{5}{ln}^{2}({m}_{e}{R}_{N})\phantom{\rule{0.16em}{0ex}}{(m{R}_{N})}^{3}m$. Special attention is paid to higher-order effects in muonic hydrogen and the related applications of various parametrizations of the proton electric form factors. We also consider the general expressions in coordinate space for the finite-size radiative contributions to the Lamb shift in two-body muonic atoms. The numerical results are present for the atoms with $Z\ensuremath{\le}10$.

We study numerically the Bloch electron wave-packet dynamics in periodic potentials to simulate laser-solid interactions. We introduce an alternative perspective in the coordinate space combined with the motion of the Bloch electron wave packets moving at group and phase velocities under the laser fields. This model interprets the origins of the two contributions (intra- and interband transitions) in the high-order harmonic generation (HHG) processes by investigating the local and global behaviours of the wave packets. It also elucidates the underlying physical picture of the HHG intensity enhancement by means of carrier-envelope phase, chirp, and inhomogeneous fields. It provides a deep insight into the emission of high-order harmonics from solids. This model is instructive for experimental measurements and provides an alternative avenue to distinguish mechanisms of the HHG from solids in different laser fields.

We examine the interaction of thermal light with matter with emphasis on two aspects that have not been considered before. By employing a fully quantized Jaynes-Cummings--type interaction model on a $V$-type three-level system, we show that multimode thermal light induces coherence in the excited material states. This is in contrast to previous studies that suggest thermal light cannot induce coherence in material systems. We also show that the ratio between the field detuning and the interaction constant has a significant influence on the characteristic time-dependent dynamics. In particular, for some ratio regimes, the thermal light induces dynamics with a ``coherentlike'' collapse and revivals pattern rather than the familiar pattern. We then extend the Jaynes-Cummings model to a two-state Born-Oppenheimer potential energy surface molecular system where the internal vibrational degrees of freedom are fully taken into account. The matter-field bipartite system is represented, and propagated, in the full electronic bond-coordinate Fock product space. We show that single-mode thermal light induces extensive excited-state vibrational coherence in the molecule that, when observed in coordinate space, exhibits wave-packet-like dynamics. The molecular Jaynes-Cummings model we propose is useful for cavity molecular dynamics simulations.

The vacuum-polarization correction for bound electrons or muons is examined. The objective is to formulate a framework for calculating the correction from bound-state quantum electrodynamics entirely in coordinate space, including the Uehling potential, which is usually isolated and treated separately. Pauli-Villars regularization is applied to the coordinate-space calculation and the most singular terms are shown to be eliminated, leaving the physical correction after charge renormalization. The conventional derivation of the Uehling potential in momentum space is reviewed and compared to the coordinate-space derivation. In atomic hydrogen, an electron bound to a proton, the largest radiative correction to the energy levels of S states is the self-energy correction, which results from the electron emitting and reabsorbing a photon. This is in contrast to muonic hydrogen, a muon bound to a proton, in which the vacuum-polarization correction dominates. The reason for the difference is that for the muon, the lighter electron-positron pair that can occur in a vacuum-polarization loop produces a large effect, while for the electron, there is no lighter particle to produce a correspondingly large correction. There has been a renewed interest in the effects of vacuum polarization due to the recent measurement of the Lamb shift in muonic hydrogen (1-3). As mentioned above, in this atom, the Lamb shift is predominantly the effect of electron vacuum polarization. However, the radius of the proton, which is deduced by comparison of the measured transition frequencies to the theoretical predictions, differs from the value obtained from spectroscopy of hydrogen and deuterium and electron scattering experiments by 7σ (4). In view of this discrepancy, a thorough review of the theory is warranted, and a number of such investigations have been carried out, with recent reviews given in Refs. (5-12). In this paper, we reexamine the theory of vacuum po- larization. The objective is to provide a formulation for the calculation of the vacuum-polarization effect entirely in coordinate space for any spherically symmetric binding field, in parallel with an earlier analysis of the self-energy correction in coordinate space (13). This provides a framework for a direct numerical evaluation of the correction for strong Coulomb fields and for non-Coulombic binding fields for which a perturbation expansion is not feasible. Vacuum polarization is particularly problematic, because it contains the most severe divergences of all the bound-state corrections. The calculation of vacuum polarization done by Wich- mann and Kroll (14) is also based on a coordinate-space formulation, but it requires the explicit analytic expression

We analyze the problem of suppression of reflections within the numerical propagation of wave packets using finite atomiclike pseudostate expansions. Artificial reflections at effective ``walls'' and reentering of probability flux into the interaction region represents a major limitation for the study of long-time evolution of atomic ionization processes. We propose two methods, the repetitive projection method (RPM) and Siegert pseudostate (SPS) propagation, and study their efficiency in suppressing reflections. It is shown that the quantum Zeno effect sets a limit on the efficiency of the RPM as well as of masking functions. For the exactly solvable propagation of a radial-free wave packet, we show that both the SPS and the RPM provide almost complete suppression of reflections without appreciable distortion in the physically relevant region of coordinate space.

A first quantized theory of the photon is developed that incorporates all of the usual rules of quantum mechanics. The state vector is argued to be proportional to the four-vector potential. Scalar products and the probability amplitude are invariant under gauge transformations that satisfy the Lorentz gauge condition. The eigenvectors of a recently constructed Hermitian position operator with commuting components provide a basis for the position representation.

A method to study weakly bound three-body quantum systems in two dimensions is formulated in coordinate space for short-range potentials. Occurrences of spatially extended structures (halos) are investigated. Borromean systems are shown to exist in two dimensions for a certain class of potentials. An extensive numerical investigation shows that a weakly bound two-body state gives rise to two weakly bound three-body states, a reminiscence of the Efimov effect in three dimensions. The properties of these two states in the weak binding limit turn out to be universal. PACS number(s): 03.65.Ge, 21.45.+v, 31.15.Ja, 02.60Nm, 9 pages, 2 postscript figures, LaTeX, epsf.sty

A general scheme of qualitative analysis is applied to molecular rovibrational problems. The classical-quantum correspondence provides a description of different classes of localized quantum rotation-vibration states associated with localized classical motion. A description of qualitative features, such as localized motion, and of qualitative changes, such as localization phenomena, is based on the concept of the simplest Hamiltonian. It uses only the topological properties of the compact reduced phase space and the action of the symmetry group on this space. The qualitative changes of the simplest Hamiltonian are analyzed as bifurcations caused by rotational or vibrational excitation. The relation between the stationary points of the classical Hamiltonian function on the reduced phase space and the principal periodic trajectories in the coordinate space is analyzed for vibrational Hamiltonians. In particular, the relation between the nonlinear normal modes, proposed by Montaldi, Roberts, and Stewart [Philos. Trans. R. Soc. London, Ser. A 325, 237 (1988)], and normal- and local-mode models widely used in molecular physics is discussed. Along with a general consideration of localized rotational and vibrational states a more detailed analysis of the vibrational dynamics of an ${X}_{3}$ molecule with the ${D}_{3h}$ symmetry, such as the ${\mathrm{H}}_{3}^{+}$ molecular ion, is given.

We consider tunneling of a nonrelativistic particle across a potential barrier. It is shown that the barrier acts as an effective beam splitter which builds up the transmitted pulse from the copies of the initial envelope shifted in the coordinate space backward relative to the free propagation. Although along each pathway causality is explicitly obeyed, in special cases reshaping can result an overall reduction of the initial envelope, accompanied by an arbitrary coordinate shift. In the case of a high barrier the delay amplitude distribution (DAD) mimics a Dirac $\ensuremath{\delta}$ function, the transmission amplitude is superoscillatory for finite momenta and tunneling leads to an accurate advancement of the (reduced) initial envelope by the barrier width. In the case of a wide barrier, initial envelope is accurately translated into the complex coordinate plane. The complex shift, given by the first moment of the DAD, accounts for both the displacement of the maximum of the transmitted probability density and the increase in its velocity. It is argued that analyzing apparent ``superluminality'' in terms of spacial displacements helps avoid contradiction associated with time parameters such as the phase time.

A full four-body implementation of the propagating exterior complex scaling (PECS) method is developed and applied to the electron-impact of helium in an S-wave model. Time-independent solutions to the Schrodinger equation are found numerically in coordinate space over a wide range of energies and used to evaluate total and differential cross sections for a complete set of three- and four-body processes with benchmark precision. With this model we demonstrate the suitability of the PECS method for the complete solution of the full electron-helium system. Here we detail the theoretical and computational development of the four-body PECS method and present results for three-body channels: single excitation and single ionization. Four-body cross sections are presented in the sequel to this article. The calculations reveal structure in the total and energy-differential single-ionization cross sections for excited-state targets that is due to interference from autoionization channels and is evident over a wide range of incident electron energies.

The time evolution of the ground state of a one-dimensional hydrogen atom in the intense laser pulse is studied directly in momentum space. A finite matrix representation of operators is developed to solve nonperturbatively the time-dependent Schr\"odinger equation. We find that some numerical limitations in coordinate space of above-threshold ionization (ATI) can be overcome, and clear physical interpretations of the numerical data are possible. Both the ATI and harmonic-generation spectra are obtained. The correlations between these two multiphoton phenomena are discussed.

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of "restricted Gaussian wave packets" consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically, and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well defined angular momentum quantum n umbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\v{c}i\v{c} et al., submitted]., Comment: 12 pages, 3 figures

A non-relativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length a is much larger than the range l of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the |a| goes to infinity limit and finite-l/a corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, r_s on the three-boson bound-state spectrum and recombination rate for |a| much greater than |r_s|. We do this by first deriving results appropriate to the strict limit |a| goes to infinity in coordinate space. We then extend these results to finite a using once-subtracted momentum-space integral equations. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance., 28 pages, 3 figures

Encouraged by the experiments on production of antihydrogen atoms in high Rydberg states we have calculated the effect of deexcitation towards lower states by a chirped train of identical unidirectional half-cycle pulses. The calculations exploit both the one-dimensional and impulse approximations providing convenient analytical formulas for the Rydberg-to-Rydberg transition amplitudes. The calculated deexcitation is shown in terms of the mean value of localization of the Rydberg wave packet in the coordinate space, the Rydberg-state population distribution, the Husimi phase-space distribution function, and the probability density distribution, each of these measures vs the length of the applied train of half-cycle pulses. The results for chirped trains are compared with those for periodic trains and examples of higher deexcitation efficiency of the chirped trains are given.

The eigenstates of nonrelativistic two-, three-, four-, and six-valence-electron atoms are classified according to their transformation properties under symmetry operations. In energy-favorable geometrical arrangements of the electrons, the spatial distribution of the wave function is constrained by the demand of invariance under rotation, inversion and permutation of particles. For a given term, the symmetry constraints may leave the energy-favorable arrangement accessible or inaccessible. In the latter case the term is not expected to belong to the low-energy part of the spectrum. Contrary, in the former case with no symmetry-induced inherent nodal surface in the multidimensional coordinate space of the multielectron wave function, the term is expected to belong to the low-energy part of the spectrum. The symmetry-based classification is confirmed by recent calculations.

We demonstrate how a general von Neumann\char21{}like measurement can be analyzed in terms of histories (paths) constructed for the measured variable $A$. The Schr\"odinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.

Ionization of hydrogen atoms by the impact of fast charged particles can be accurately treated theoretically using simple two-center wave functions to describe the scattering system both initially and finally. For the final state, we use a continuum distorted wave (CDW) that contains the product of three Coulombic distortion factors (one for each two-body interaction), hence it is called the 3C wave function. This CDW (3C) wave function is ideal for studying fast collisions since it is asymptotically correct in all asymptotic domains of momentum space for all configurations of the three particles in coordinate space [S. Jones and D. H. Madison, Phys. Rev. A $62,$ 42 701 (2000)]. Coulomb distortion in the entrance channel is provided by an eikonal initial state (EIS), and this CDW-EIS approximation, introduced by Crothers and McCann [J. Phys. B $16,$ 3229 (1983)], has proven to be the most accurate perturbative method ever devised within a two-state approximation. The first fully differential cross sections for ion-atom ionization in the CDW-EIS approximation are presented. In addition, by considering projectiles of different mass and charge, it will be shown that although the charge of the projectile is important, the mass of the projectile plays virtually no role in the vast majority of fast ionizing collisions.

This paper reports a method to build a semiclassical wave function corresponding to an invariant torus satisfying Einstein-Brillouin-Keller quantization conditions. Instead of calculating the stability matrix of the trajectory at each step, as in the standard method of Keller [Ann. Phys. (N.Y.) 4, 180 (1958)] or the modification of Maslov and Fedoriuk [Semiclassical Approximations in Quantum Mechanics (Reidel, Dordrecht, 1981)], we use the actual density of the trajectory, calculated by running the trajectory and counting passages through cells in coordinate space. The method is tested for a system of coupled Morse oscillators, and found to be comparable in accuracy to the standard method. It may be more useful than the standard method for testing ideas for semiclassical quantization in the chaotic regime. (c) 2000 The American Physical Society.

The local properties and accuracy of the positronium negative-ion $({\mathrm{Ps}}^{\ensuremath{-}})$ ground-state wave functions obtained by the stochastic variational method (SVM) and by direct solution of the Schr\"odinger equation with the help of the correlation-function hyperspherical-harmonic method (CFHHM) are studied and compared. Though the energy, calculated by both methods, agrees to up to ten digits, the amplitudes of the values of the operator $D=H\ensuremath{\Psi}/E\ensuremath{\Psi}\ensuremath{-}1,$ characterizing local deviation of the wave function from its true value, in all of the coordinate space in the SVM are consistently larger (by up to five orders of magnitude) than in the CFHHM, despite the fact that the SVM observables except $〈\ensuremath{\delta}({\mathbf{r}}_{k})〉$ converge to significantly more digits than the CFHHM observables for their respective selected bases.

Low-energy muon-transfer cross sections and rates in collisions of muonic atoms with hydrogen isotopes are calculated using a six-state close-coupling approximation to coordinate-space Faddeev-Hahn-type equations. In the muonic case satisfactory results are obtained for all hydrogen isotopes and the experimentaly observed strong isotopic dependence of transfer rates is also reproduced. A comparison with results of other theoretical and available experimental works is presented. The present model also leads to good transfer cross sections in the well-understood problem of antihydrogen formation in antiproton-positronium collision., Comment: 18 pages REVTeX, accepted for publication in Phys. Rev. A

A Green's-function approach is used to solve the Schroedinger equation in an effective potential (V/sub 0/), which is the sum of independent-atom static potentials. The equation for the Green's function is conveniently solved in momentum space (MS), where the MS ''potentials'' (Fourier transforms of the atom-centered potentials) have translational symmetry. The Green's function is then used to construct the solution to the Schroedinger equation for scattering in the potential V-V/sub 0/ (where V is the e-molecule static potential plus a local exchange potential) relative to scattering in V/sub 0/. This solution is found in coordinate space using single-center expansions about the internuclear midpoint. These are more rapidly convergent for V-V/sub 0/ than for V or V/sub 0/ alone. The sum of the amplitudes for scattering in V/sub 0/ and in V-V/sub 0/ relative to V/sub 0/ then represents the amplitude for scattering from the molecule. This method is intended to combine the dynamical methods best suited for each type of potential (multicenter for V/sub 0/ and single center for V-V/sub 0/). It also exposes the shortcomings of the use of V/sub 0/ alone.

Calculations of ionization cross sections have been performed for removing electrons from $2s$ and $2p$ levels in hydrogenlike atoms by heavy-charged-particle impact using the binary-encounter approximation (BEA). The ratio of $\frac{{\ensuremath{\sigma}}_{{L}_{\mathrm{II}}}}{{\ensuremath{\sigma}}_{{L}_{\mathrm{I}}}}$ is predicted to peak when the projectile has a velocity equal to 0.3 times the velocity of the atomic electron. Experimental values of $\frac{{\ensuremath{\sigma}}_{{L}_{\mathrm{II}}}}{{\ensuremath{\sigma}}_{{L}_{\mathrm{I}}}}$, in agreement with Born predictions, peak at a velocity significantly lower than that predicted by BEA calculations. By transforming to coordinate space in an approximate way, predictions for the impact-parameter dependence are made. In the $2s$ case, the ionization probability has a pronounced local minimum for impact parameters near half the atomic radius, corresponding to a node in the $2s$ wave function. The ionization probabilities for arbitrary charged particles on arbitrary targets may be easily found by applying scaling laws to values tabulated over a wide range of projectile velocities.

This paper derives a number of sum rules for nonrelativistic three-body scattering. These rules are valid for any finite region ..sigma.. in the six-dimensional coordinate space. They relate energy moments of the trace of the onshell time-delay operator to the energy-weighted probability for finding the three-body bound-state wave functions in the region ..sigma... If ..sigma.. is all of the six-dimensional space, the global form of the sum rules is obtained. In this form the rules constitute higher-order Levinson's theorems for the three-body problem. Finally, the sum rules are extended to allow the energy momtns have complex powers.

A six-dimensional integral in the coordinate space of two particles 1 and 2, where the integrand contains Coulomb functions for both the particles, the term 1/ vertical-bar r$sub 1$ - r$sub 2$vertical-bar, and an exponential term, is reduced to a double integral in real space. The resulting integral can finally be integrated numerically with a high degree of accuracy. This technique has been verified by its application to the Nordsieck integral for bremsstrahlung with different sets of parameters and compared with the exact method of Nordsieck. The error estimate is always found to be less than 0.01%.

The $〈{r}^{n}〉$ expectation values have been extracted from the known electron momentum distributions for hydrogen, helium, argon, and krypton atoms with the use of a semiclassical approach. These values compare fairly well with their Hartree-Fock counterparts. This procedure provides a link between the distribution of electrons in the momentum space and that in the coordinate space.