The semiclassical WKB approximation method is reexamined in the context of nonrelativistic quantum-mechanical bound-state problems with broken supersymmetry (SUSY). This gives rise to an alternative quantization condition (denoted by BSWKB) which is different from the standard WKB formula and also different from the previously studied supersymmetric (SWKB) formula for unbroken SUSY. It is shown that to leading order in ħ, the BSWKB condition yields exact energy eigenvalues for shape-invariant potentials with broken SUSY (harmonic oscillator, Pöschl-Teller I and II) which are known to be analytically solvable. Further, we show explicitly that the higher-order corrections to these energy eigenvalues, up to sixth order in ħ, vanish identically. We also consider a number of examples of potentials with broken supersymmetry that are not analytically solvable. In particular, for the broken SUSY superpotential W=${\mathit{Ax}}^{2\mathit{d}}$ [A>0, d=(integer)], we evaluate contributions up to the sixth order and show that these results are in excellent agreement with numerical solutions of the Schrödinger equation. While the numerical BSWKB results in lowest order are not always better than the corresponding WKB results, they are still a considerable improvement because they guarantee equality of the corresponding energy eigenvalues for the supersymmetric partner potentials ${\mathit{V}}_{+}$ and ${\mathit{V}}_{\mathrm{-}}$. This is of special importance in those situations where these partner potentials are not related by parity.

We extend the interference in the phase-space algorithm of Wheeler and Schleich [W. P. Schleich and J. A. Wheeler, Nature 326, 574 (1987)] to the case of a compact, spherical topology in order to discuss the large j limits of the angular momentum marginal probability distributions. These distributions are given in terms of the standard rotation matrices. It is shown that the asymptotic distributions are given very simply by areas of overlap in the classical spherical phase-space parametrized by the components of angular momentum. The results indicate the very general validity of the interference in phase-space concept for computing semiclassical limits in quantum mechanics.

We review work on constrained systems in which (3+1)-dimensional field theories are reduced to effective (2+1)-dimensional ones. Known results are extended to encompass the Dirac equation and the nonrelativistic limit is examined. We discuss to what extent this system can really be made two dimensional and obtain a lower bound on the thickness. Some comments are made about recent theories involving fractional statistics.

We study the classical limit of the equal-time relativistic quantum transport theory. We discuss in qualitative terms the need to fold first the Wigner function with a coarse-graining function. Only then does the singularity at ħ→0 seem to be manageable. In the limit ħ→0, we obtain the relativistic Vlasov equations for the particle and the antiparticle sector of the Fock space. Similarly, we address the evolution equations of the spin and the magnetic-moment density.

The final result of a 2s ${}_{}{}^2{}_{}{}^{}$${\mathit{S}}_{1/2–}$2p ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{1/2}$ Lamb-shift measurement by the laser resonance method in hydrogenic phosphorus is reported. Metastable 2s ${}_{}{}^2{}_{}{}^{}$${\mathit{S}}_{1/2}$ ions were prepared using a β≊0.08 velocity beam of ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}_{}^{14+}{}_{}{}^{}$ ions obtained from the MP tandem accelerator at the Centre de Recherches Nucléaires in Strasbourg. From the metastable state the transition to the 2p ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{3/2}$ state was induced with a high-power dye laser leading to a measured transition energy of ΔE=2.231 33(12) eV. Subtracting the well-known fine-structure splitting Δ${\mathit{E}}_{\mathrm{FS}}$=2.314 82(2) eV, the Lamb shift is deduced to be ${\mathit{E}}_{\mathrm{expt}}$(LS)=20 188(29) GHz. This result is compared with the theoretical value ${\mathit{E}}_{\mathrm{theor}}$(LS)=20 254(10) GHz and with other experimental results for the Lamb shift in the region Z≤18. Finally, the utility of the Lamb-shift measurements in testing the G(Zα) function of the self-energy, which includes only terms in (Zα${)}^{\mathit{n}}$, n≥6, is discussed.

We derive, to leading order, the momentum-space functionals of the exchange energy and of the quantum corrections to the electrostatic energy. The energy functionals thus refined are used to introduce a Kohn-Sham scheme for self-consistent momentum-space calculations. This scheme is applied to helium and beryllium. The results differ from those obtained in the standard Kohn-Sham scheme in configuration space. The total binding energies predicted by the momentum-space computation are somewhat better than those of the configuration-space method.

For a four-particle system consisting of two electrons and two positrons, the nonadiabatic wave function is constructed with the use of an expansion in terms of explicitly correlated Gaussian-type basis functions and a Cartesian-coordinate laboratory frame. Motions of all particles are correlated at the same time in the wave function. The energy of the center-of-mass motion is effectively eliminated from the total nonrelativistic energy of the system by defining the variational principle based on the internal Hamiltonian. The ground-state energy is computed for different lengths of Gaussian expansions and values are compared with previous literature results. Our best estimation of the binding energy of the positronium molecule is 0.435 eV.

A subnatural linewidth measurement of magnetic-dipole and electric-quadrupole hyperfine coupling constants in the atomic Na 3p ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{3/2}$ level are reported. The measurements are made using a pump–delayed-probe method, with delays ranging to 135 ns, more than eight times the natural lifetime of the 3p ${}_{}{}^2{}_{}{}^{}$${\mathit{P}}_{3/2}$ level. A measurement of hyperfine-induced beats in the time evolution of the electronic alignment permitted determination of the hyperfine frequencies to well beyond the natural width in that level. Hyperfine coupling constants A=18.534(15) MHz and B=2.724(30) MHz are obtained, these being more than 2 times more precise than any previous determination of those quantities.

The finite-element method provides a convenient and accurate procedure for the calculation of the expectation values of quantum observables. We calculated energies, wave functions, and expectation values of ${\mathit{r}}_1^{\mathit{n}}$ for n=-1, 1, and 2, and of πδ(${\mathbf{r}}_1$) for the singlet n ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ and triplet n ${}_{}{}^3{}_{}{}^{}\mathrm{S}$ states (n=1,2,3,4) of helium. In contrast to the standard methods with globally defined basis functions, the accuracy of the expectation values of physical observables is comparable to the accuracy of the eigenvalues. The results are reported here and compared with those of Baker et al. 2 [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); Phys. Rev. A 41, 1247 (1990)], Drake [Nucl. Instrum. Methods Phys. Res. B 31, 7 (1988)], Pekeris [Phys. Rev. 115, 1216 (1959)], Accad et al. [Phys. Rev. A 4, 516 (1971)], and Haftel and Mandelzweig [Phys. Rev. A 38, 5995 (1988)].

The use of vector recursion relations for calculating successive approximants to banded secular determinants permits a considerable enlargement of the class of potentials whose energy eigenvalues can be evaluated to high precision by the method of Hill determinants. To demonstrate this, we study the rational potential V(x)=${\mathit{x}}^2$+λ${\mathit{x}}^2$/(1+${\mathit{gx}}^2$) in one dimension for g>0. Using operator methods and Hill determinants in conjunction with vector recursion relations, we display the convenience with which the energy eigenvalues of the problem can be calculated to great accuracy for positive as well as negative values of λ. The results for λ<0 have not previously been obtained. The energy eigenvalue spectrum of this potential is also shown to possess discontinuities when the coupling parameters λ and g take suitable limiting values, pointing to the need for exercising care in the construction of perturbative solutions to the problem.

We report a measurement of the lifetime of the 1s2s ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$ state in two-electron ${\mathrm{Br}}^{33+}$. This state decays to the 1${\mathit{s}}^2$ ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$ ground state only by the simultaneous emission of two photons. Ions in the 1s2s ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$ level are produced by excitation of energetic bromine ions in a thin carbon foil. The lifetime is determined by measuring the change in the rate of two-photon coincidences as a function of the foil-detector separation. The lifetime measurement yields 39.32(32) ps in agreement with a theoretical value of 39.63(16) ps.

The kinetic-energy difference ΔT=T-${\mathit{T}}_{\mathit{s}}$[${\mathrm{\rho }}_0$] is calculated for the helium isoelectronic series and for the beryllium atom. ${\mathit{T}}_{\mathit{s}}$[${\mathrm{\rho }}_0$] is in this case the kinetic energy corresponding to a noninteracting N-particle system which, however, has the same density ${\mathrm{\rho }}_0$ as the exact interacting system. These densities ${\mathrm{\rho }}_0$ were assumed in the present case to be well represented by those coming from the optimal Hylleraas-type expansions for the He isoelectronic series and by the Bunge-Esquivel 650-term configuration-interaction wave function for Be. The calculations are carried out by means of a constrained variational method based on local-scaling transformations. The connection between this approach and the one based on the Kohn-Sham equations is discussed.

The energies and wave functions of the 1${\mathit{s}}^2$2snl states of beryllium are calculated with a full-core plus correlation method. Eight excited states (2p ${}_{}{}^{1,3}{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, 3s ${}_{}{}^{1,3}{}_{}{}^{}$S, 3p ${}_{}{}^{1,3}{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, and 3d ${}_{}{}^{1,3}{}_{}{}^{}$D) are studied. A restricted variational method is used to extrapolate a better nonrelativistic energy. The relativistic corrections are calculated with first-order perturbation theory. The calculated excitation energies (relative to the ground state) are compared with experiment. For the 2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, 3s ${}_{}{}^{1,3}{}_{}{}^{}$S, 3p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, and 3d ${}_{}{}^3{}_{}{}^{}$D states, the predicted energies agree with experiment to about 1 ${\mathrm{cm}}^{\mathrm{-}1}$. However, the discrepancies are larger for 2p ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, 3p ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}^{\mathit{o}}$, and 3d ${}_{}{}^1{}_{}{}^{}$D. The relativistic corrections are found to be critically important in these comparisons. The predicted fine-structure splittings for 2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_{2,1,0}^{\mathit{o}}$ are 2.360 and 0.637 ${\mathrm{cm}}^{\mathrm{-}1}$. They agree well with the 2.35 and 0.64 ${\mathrm{cm}}^{\mathrm{-}1}$ in the experiment. The predicted ${}_{}{}^{10}{}_{}{}^{}–_{}^9{}_{}{}^{}$Be 2s ${}_{}{}^1{}_{}{}^{}$S–3d ${}_{}{}^1{}_{}{}^{}$D isotope shift is 14.08 GHz. This also agrees with the 14.05(4) GHz in the experiment. The lifetime of the 1${\mathit{s}}^2$2s2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_1^{\mathit{o}}$ is calculated using the intermediate-coupling scheme.

Single- and double-photoionization processes of gaseous nitric oxide (NO) have been studied in the photon energy region 37–100 eV by use of time-of-flight mass spectrometry and a photoion-photoion-coincidence method together with synchrotron radiation. The single- (${\mathrm{\sigma }}^{+}$) and double- (${\mathrm{\sigma }}^{2+}$) photoionization cross sections of NO are determined. Dissociation ratios of the precursors ${\mathrm{NO}}^{+}$ and ${\mathrm{NO}}^{2+}$ are determined separately at the excitation energies where the single and double photoionization are in competition. The results show that the metastable ${\mathrm{NO}}^{2+}$ ions amount to about 15% at higher photon energies in the double-photoionization processes. Ionic branching ratios and the partial cross sections for the individual ions produced from the precursors ${\mathrm{NO}}^{+}$ and ${\mathrm{NO}}^{2+}$ are also presented. The threshold for the dissociative double photoionization was found to be 41.0±0.5 eV.

We have determined the magnetic-substate alignment produced in double-electron capture by the H-like projectiles ${\mathrm{C}}^{5+}$ and ${\mathrm{B}}^{4+}$ from He atoms by measuring the anisotropy of the Auger electron emission from the doubly excited 1s2l2l’ ${}_{}{}^2{}_{}{}^{}$L projectile states formed in the collision. This work includes additional measurements and expands upon our previous short communication on the (${\mathrm{C}}^{5+}$,He) system [Phys. Rev. A 43, 607 (1991)]. The measurements yield ${\mathit{M}}_{\mathit{L}}$ substate population fractions which challenge theory at nearly the finest quantum state detail. We observe large differences between the results for ${\mathrm{B}}^{4+}$ and ${\mathrm{C}}^{5+}$ over the velocity range 0.25–0.50 a.u. for the substate and total 1s2l2l’ ${}_{}{}^2{}_{}{}^{}$L relative cross sections. Substantial population of ${\mathit{M}}_{\mathit{L}}$>0 states shows the importance of rotational coupling in these slow collisions. The large anisotropies observed demonstrate that substantial errors can result from inferring ‘‘cross sections’’ from single-angle measurements and the assumption of isotropic electron emission.

Relativistic distorted-wave cross sections are calculated for electron-impact ionization from the 21 subshells with n=3, 4, and 5 in several types of ions. Based on these results and the general, but very simple form for the cross section derived by Sampson and Zhang [Phys. Rev. A 45, 1657 (1992)] a very rapid, but quite accurate method using fits to the reduced cross sections is given. This applies to highly charged ions with any degree of complexity and should be very convenient for use in plasma modeling. The procedure is illustrated by application to inner-shell ionization of Cu-like gold ions, where good agreement is obtained with detailed direct relativistic distorted-wave calculations.

The metastable Be anion has been experimentally investigated using a crossed laser–negative-ion-beam apparatus. The Be(2 ${}_{}{}^3{}_{}{}^{}$P) electron affinity of 261±10 meV has been determined by photodetached-electron spectroscopy. It is also demonstrated that the Be(2 ${}_{}{}^3{}_{}{}^{}$P) excitation energy of 2.723±0.007 eV can be determined by means of a hybrid technique involving the use of both autodetached- and photodetached-electron spectroscopies.

Extensive experimental studies of Mg-Ne and Mg-Ar fractional collisions are reported. In the fractional-collision process, two-color, two-photon absorption takes place during a single binary collision. Polarization and relative intensity spectra were measured in an approximately ±50-${\mathrm{cm}}^{\mathrm{-}1}$ range around the atomic Mg 3s ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$→3p ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}_1$→5s ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$ two-photon resonance. The generally-weak-field spectra show a remarkable phenomenology associated with the relative linear-polarization directions of the two exciting laser beams and with their separate detunings from atomic one-photon resonance. The polarization and relative absorption spectra display effects due to relative detuning, multiple Condon points, and laser intensity. Details of our experimental approach, and spectra for the process in Mg-Ar and Mg-Ne are presented and discussed. The data are seen to contain information on the short-range collision dynamics of the Mg–rare-gas pairs, and on the development of the alignment over the course of the collision.

Aside from empirical formulas, the radiative-recombination cross section and coefficient are usually given in tabulated forms instead of analytic formulas. Here, we give analytic expressions in the form of expansions for the recombination cross section as a function of the electron energy E for low E, and for the recombination coefficient as a function of the temperature T for low T. The expansion coefficients are combinations of confluent hypergeometric functions, and are tabulated for a large number of the final principal and angular-momentum quantum numbers n and l. It is shown that the recombination cross section for arbitrary nuclear charge number Z is independent of Z, while the recombination coefficient for T/${\mathit{Z}}^2$≪1.58×${10}^5$ K increases as ${\mathit{Z}}^2$. Excellent agreement is found with the published tabulated values.

The cross section for the production of ${\mathit{e}}^{+}$-${\mathit{e}}^{\mathrm{-}}$ (K orbit) pairs by very-high-energy fully stripped heavy ions is calculated for large impact parameters, where perturbation theory is valid. To achieve accuracy a large range of angular momenta and energies is explicitly computed. The remaining high-angular-momentum and high-energy domain are included by means of a modified version of an asymptotic expansion devised by Hall [Phys. Rev. 45, 620 (1934); Rev. Mod. Phys. 8, 358 (1936)]. An approximation to the small-impact-parameter, nonperturbative contribution is taken from previous work to obtain a reliable estimate for the practically important ${\mathrm{Au}}^{79+}$ on ${\mathrm{Au}}^{79+}$ cross section.