Statistical mechanics is applied to study the generalization properties of a two-layer neural network trained to implement a linearly separable problem. For a stochastic learing algorithm the generalization error as a function of the training set size is calculated exactly. The network with three hidden units experiences two first-order phase transitions due to an asymmetric freezing of the hidden units. Compared to a simple perceptron the committee machine is found to generalize worse.

The Josephson relation V=nhν/2e has been verified experimentally to 3 parts in ${10}^{19}$ [A. K. Jain, J. E. Lukens, and J.-S. Tsai, Phys. Rev. Lett. 58, 1165 (1987)]. Motivated by this result, we propose a differential-frequency-readout system by two sets of hysteretic Josephson junctions rf biased at millimeter wavelengths. Because of the Josephson relation, the proposed differential-frequency-readout system is not limited by photon fluctuation, which limits most photon-detection schemes. In the context of the Stewart-McCumber model [W. C. Stewart, Appl. Phys. Lett. 12, 277 (1968); D. E. McCumber, J. Appl. Phys. 39, 3113 (1968)] of Josephson junctions, we show theoretically that the differential frequency of the two milliwave biases can be read out by the proposed system to unprecedented accuracy. The stability of the readout scheme is also discussed. The measurement uncertainty of the readout system resulting from the intrinsic thermal noise in the hysteretic junctions is shown to be insignificant. The study of two single junctions can be extended to two sets of Josephson junctions connected in series (series array) in this measurement scheme provided that junctions are separated by at least 10 μm [D. W. Jillie, J. E. Lukens, and Y. H. Kao, Phys. Rev. Lett. 38, 915 (1977)]. The sensitivity for the differential frequency detection may be increased by biasing both series arrays to a higher constant-voltage step.

In this paper, the Lewis-Riesenfeld invariant theory is generalized for the study of systems with non-Hermitian time-dependent Hamiltonians. It is then used to study the nonadiabatic cyclic evolution and the Aharonov-Anandan phase. It is shown that the study of noncyclic evolution can be reduced to the study of cyclic evolution. The two-level dissipative system and the classical time-dependent harmonic oscillator are discussed as illustrative examples.

It is shown how the Zakharov-Shabat (ZS) eigenvalue problem for rational reflection coefficient may be reduced to the ZS problem with zero reflection coefficient. The soliton solutions to this reduced problem are obtained using the Bäcklund transform. Hence the solutions to the original problem are shown to be half solitons. It is demonstrated how selective pulses in nuclear magnetic resonance may be calculated using this technique. In particular, almost perfect 90° self-refocused and 180° refocusing selective pulses are demonstrated.

The consequences of quantizing the Fokker-Wheeler-Feynman model of electrodynamics, treating the Lagrangian via its acceleration-dependent (1/c) power-series representation, is examined using recently validated methods. An exact treatment of this acceleration dependence yields, under certain circumstances, high-energy resonant modes. In the past, such modes have been assumed unphysical and have been removed by perturbative or order-reduction techniques. However, these modes appear to be of physical significance. This conclusion follows because this completely ab initio calculation, with no adjustable parameters, has a number of successes. It provides a description for resonances observed in the electron-positron emission from heavy-ion collisions, in particular, and in diproton collisions and, possibly, in other collision experiments as well.

A local hidden-variable model is exhibited for the experiments by Aspect, Grangier, and Roger [Phys. Rev. Lett. 47, 460 (1981); 49, 91 (1982)] and Aspect, Dalibard, and Roger [Phys. Rev. Lett. 49, 1804 (1982)] measuring polarization correlation of optical-photon pairs. The model agrees with quantum-mechanical predictions for all measurable quantities even with ideal polarizers and detectors, and emphasizes the need of a high degree of directional correlation, besides the correlation of spin (or polarization or other quantities), in any test of locality. It is proved that homogeneous inequalities, involving only coincidence detection rates, cannot discriminate between quantum mechanics and local theories, which invalidates all previously used empirical tests. The role of supplementary assumptions, like the so-called no enhancement, for the derivation of Bell’s inequalities is discussed. Finally it is conjectured that quantum mechanics might be compatible with local realism, if we assume that not all self-adjoint operators represent observables and not all density operators represent states.

In this paper a relativistic quantum theory introduced by de Groot and Ruijgrok [Nucl. Phys. 101B, 95 (1975)] is reformulated. It is shown that it has the correct nonrelativistic limit and that it gives the same lowest-order relativistic corrections as the Breit equation. In this paper it is applied to bound systems of two charged particles with spin 1/2. For small coupling constants the lowest-order fine structure (up to order ${\mathrm{\alpha }}^4$) for all states and all mass ratios is calculated. No expansion in the mass ratio is made, so that the full recoil effects are obtained. It embraces as special cases the well-established fine structure of the hydrogen atom and positronium and the Fermi splitting of muonium. For the strong-coupling case the critical couplings are calculated. The convergence to the static limit for couplings in the neighborhood of the critical point is extremely slow. The bound-state masses for states with the same total angular momentum J but opposite parity are not equal at the critical point, so there is no parity doubling. The results are compared with a number of other theories.

Beginning with the quaternionic generalization of the Schrödinger equation, we present a simple system consisting of two successive square barriers of general (quaternionic) height, and discuss situations in which the predictions of quaternionic quantum mechanics differ from those of the usual complex theory. Specifically, we show that the transmission coefficient may pick up a phase change on reversal of the barrier order. We comment on why this phase change would not necessarily be observed in experiments. Finally, we present a necessary condition for the magnitude of the transmission coefficient to change under reversal of the direction of traversal of the barrier.

We report measurements of excited states in the three stable isotopic variants of molecular hydrogen: ${\mathrm{H}}_2$, ${\mathrm{D}}_2$, and HD. The lowest rovibrational levels of the ${\mathit{EF}}^1$${\mathrm{\Sigma }}_{\mathit{g}}^{+}$ state have been measured by Doppler-free two-photon laser spectroscopy for all three molecules. The accuracy is 0.015 ${\mathrm{cm}}^{\mathrm{-}1}$, and will improve to 0.003 ${\mathrm{cm}}^{\mathrm{-}1}$ when wavelength calibrations of ${\mathrm{I}}_2$ reference transitions are completed. We also describe accurate measurements of high Rydberg np states in HD, and their analysis using multichannel quantum-defect theory. For HD these measurements give a result of 124 568.479(20) ${\mathrm{cm}}^{\mathrm{-}1}$ for the ionization potential, in excellent agreement with theory. For ${\mathrm{H}}_2$, the measurements of EF-state transitions give results slightly lower than previous experimental values, but the ionization potential remains in good agreement with theory. For ${\mathrm{D}}_2$ the results reported here, when taken in combination with recent measurements of transitions between excited states by Jungen et al. [J. Mol. Spectrosc. (to be published)], yield a value for ionization potential that is again quite close to the theoretical result.

We report a compilation of Roothaan-Hartree-Fock (RHF) wave functions for the ground states of He through Xe, with atomic orbitals expressed in terms of Slater-type functions. Slight increases in the size of the basis set with respect to those used in Clementi and Roetti’s tables [At. Data Nucl. Data Tables 14, 177 (1974)] turn out to yield total energies to better than eight figures, reducing between 21 and 2770 times the energy errors in the above tables, and also improving over the previous numerical HF energies computed by Froese Fischer $[—The Hartree-Fock Method for Atoms (Wiley, New York, 1977)]. We also report 10-digit numerical HF energies with different results for Cr, Cu, Nb, Mo, Ru, Rh, Pd, and Ag.

We discuss the binding of three unit charges ${\mathit{q}}_{\mathit{i}}$=±1,∓1,∓1, with various constituent masses ${\mathit{m}}_{\mathit{i}}$. It is found that the most convenient variables are the normalized inverse masses defined by ${\mathrm{\alpha }}_{\mathit{i}}$=${\mathit{m}}_{\mathit{i}}^{\mathrm{-}1}$/(${\mathit{m}}_1^{\mathrm{-}1}$+${\mathit{m}}_2^{\mathrm{-}1}$+${\mathit{m}}_3^{\mathrm{-}1}$). The region where this three-body system is stable against dissociation into a neutral atom and an isolated charge is restricted to a narrow band that is characterized by small values of ‖${\mathrm{\alpha }}_2$-${\mathrm{\alpha }}_3$‖ and includes the well-known cases ${\mathrm{H}}_2^{+}$, ${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{-}}$${\mathit{e}}^{\mathrm{-}}$, and ${\mathrm{H}}^{\mathrm{-}}$. Some properties of this region of stability are presented. A method of proof of instability of ${\mathit{e}}^{\mathrm{-}}$${\mathit{pe}}^{+}$ is proposed.

Elaborate multiconfiguration Hartree-Fock wave functions have been obtained for the 4s ${}_{}{}^2{}_{}{}^{}$S, 3d ${}_{}{}^2{}_{}{}^{}$D, and 4p ${}_{}{}^2{}_{}{}^{}$P° states in ${\mathrm{Ca}}^{+}$ in order to investigate the core-valence correlation effects on the transition probabilities of the electric dipole (E1) and quadrupole (E2) processes involving these levels. In parallel, model-potential calculations using a core-polarization potential have been performed. They support the large reduction of the oscillator strengths found in the ab initio approach for both the E1 and E2 processes. The agreement between the two sets of results is satisfactory only if the polarization term is included in the Hartree-Fock variational procedure. The present theoretical ${}_{}{}^2{}_{}{}^{}$P${\mathrm{°}}_{1/2,3/2}$ lifetime values differ by 3.5% from the most recent fast-beam laser measurements with an estimated single-standard-deviation uncertainty of about 1%.

We assume that the nucleus is a relativistic point mass with no internal degrees of freedom and derive an expression for its motional contribution to the energies of atomic levels in jj coupling. Our formalism has been implemented in a computer program module sms used in conjunction with the grasp2 multiconfiguration Dirac-Fock package. Representative calculations are presented for some atoms and ions.

Excitation energies and oscillator strengths from the ${}_{}{}^1{}_{}{}^{}$${\mathit{S}}_0$ ground state to the first ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_1$ and ${}_{}{}^1{}_{}{}^{}$${\mathit{P}}_1$ excited states of Cd-like ions are calculated by using the multiconfiguration relativistic random-phase approximation. Results are compared with other theories and with experiment. The discrepancy between the theoretical and experimental results is discussed.

Optical isotope shifts were measured on the 618-nm, ground-state transition of Hf i for the isotopes ${}_{}{}^{174\mathrm{-}180}{}_{}{}^{}\mathrm{Hf}$. For the isotope pair 178-180, a specific mass shift of approximately -8 times the normal mass shift was deduced for this transition. The large size of this shift is consistent with other specific mass shifts that result from a broken d-shell electron pair. A transtion electronic factor of ${\mathit{E}}_{618}$=0.158±0.012 was extracted from the data. Excited-state hyperfine coefficients for the odd isotopes ${}_{}{}^{179}{}_{}{}^{}\mathrm{Hf}$ and ${}_{}{}^{177}{}_{}{}^{}\mathrm{Hf}$ were also deduced. The values of mean-square-charge radii obtained for the isotopic chain agree well with previously published results.

Nuclei are frequently assumed to be spherically symmetric distributions of (positive) charge in atomic calculations. The Fermi distribution function ${\mathrm{\rho }}_0$/(1+${\mathit{e}}^{\left(\mathit{r}\mathrm{-}\mathit{c}\right)/\mathit{a}}$) models the nuclear ‘‘skin’’ in addition to the finite extension of the nuclear charge distribution. It is a popular choice when finite-difference methods are used in calculations of atomic structure because all derivatives of the corresponding potential-energy function are finite and continuous in the region 0<r<∞ when a>0. We investigate such ‘‘Fermi nuclei’’ in the context of relativistic basis-set calculations: We present upper limits on the Dirac-Fock and Dirac-Fock-Breit ground-state energies of heliumlike ions with Z=10,20,...,160; we also present a basis-set estimate of the Dirac-Fock energy of the neutral mercury atom.

A formula relating the dipole polarizability and the exchange energy is derived for alkali-metal dimer cations. The theory is based on the fact that the unsymmetrized polarized wave function gives both the polarizability and the exchange energy through the Holstein-Herring integral. The results are seen to be remarkably accurate for large interatomic distances.

A method is introduced that allows the determination of the Kohn-Sham effective potential and the corresponding Kohn-Sham orbitals belonging to an arbitrary electron density with boundary conditions of a bound and finite system. For atoms or molecules the exchange-correlation potential and the exchange-correlation energy can be obtained also. The procedure easily can be implemented using existing density-functional programs.

The previously reported hyperspherical close-coupling method is extended to treat dipole transitions. In so doing, we introduce the dipole density function, which permits us to visualize various components of the dipole matrix as a function of the hyperradius and to study the convergence rate of the length and acceleration forms. In the adiabatic approximation, the acceleration form is shown to be always inferior to the length form regardless of the quality of the adiabatic wave function. We present detailed comparisons with other high-precision theoretical results for the bound-bound process He(1sns)+hν→He(1smp). Excellent agreement between theory and experiment is attained for the photoionization of He above ${\mathrm{He}}^{+}$(1s) threshold.

We describe a practical, rigorous, QED procedure for performing accurate calculations of the screened self-energy for high-Z ions based on a realistic local atomic potential. Using a basis-set algorithm, we show that the procedure involves a rapidly convergent partial-wave expansion that, in contrast to previous methods, requires explicit computation of only the first few partial waves. We apply the method to the 2s-2${\mathit{p}}_{1/2}$ transition of Li-like U, the 3s-3${\mathit{p}}_{3/2}$ transition of Na-like Pt, and the 4s-4${\mathit{p}}_{1/2}$ transition in Cu-like Bi. When combined with calculations of the screened Uehling vacuum-polarization term and earlier correlation calculations, the results agree with experiment to within several tenths of an eV.