Analytic expressions are obtained for direct and exchange penetration energies, corrections to polarization energies, and nonadiabatic corrections in high-angular-momentum Rydberg states. It is found that exchange energies and corrections to polarization energies are of comparable magnitude to that of the direct term and must be included in the analysis of these states.

Phase-integral formulas obtained by Fröman, Fröman, and the present author for the limit of $u_N$(l,E,r)/$r^{l+1\mathrm{}}$ as r→0, where $u_N$(l,E,r) is a normalized nonrelativistic wave function for bound or unbound states, are compared with corresponding formulas obtained by Pratt and co-workers from analytical perturbation theory. It is demonstrated that the phase-integral formulas are practically always advantageous to the perturbation-theory formulas. If one substitutes the perturbation-theory approximation for the energy into the respective phase-integral formula for the limit of $u_N$/$r^{l+1\mathrm{}}$ as r→0 and performs approximations according to a well-defined scheme, the approximate perturbation-theory formulas for the limit of $u_N$/$r^{l+1\mathrm{}}$ as r→0 are obtained. For bound s states the phase-integral formula and the perturbation-theory formula coincide with the Fermi-Segrè formula in the nonrelativistic approximation.

We present a discrete basis-set technique for calculating Bethe logarithms for eigenstates of the hydrogen atom. Although our initial results for the 1s and 2s states are accurate to only about 2%, a high-order Neville-Richardson extrapolation yields Bethe logarithms accurate to better than 1 part in ${10}^{13}$, some three digits more precise than the best previous calculation. The dramatic improvement in accuracy illustrates the power of Neville-Richardson extrapolation, which may well be found useful for improving the accuracy of numerical calculations in other areas of atomic physics.

The theory of radiative processes in quantum theory is formulated on the basis of self-energy, in analogy to classical radiation theory, and is explicitly carried out for the calculation of the Lamb shift and spontaneous emission.

Projection operators describing electron scattering from multielectron atoms (and ions) have recently been derived in detail [A. Temkin and A. K. Bhatia, Phys. Rev. A 31, 1259 (1985)]. The operators (P and Q) involve the eigenspectrum of an auxiliary eigenvalue equation in addition to the eigenfunctions of the target system. In the present paper, a sum rule for these auxiliary eigenvalues is derived. It is verified for the two-electron target with use of an open-shell approximation both in the elastic and inelastic domains. For Hylleraas target functions the auxiliary eigenspectrum is infinite; nevertheless, a variational solution of the auxiliary problem corresponding to a simple but judiciously chosen Hylleraas target approximation is shown to exhaust the sum rule to better than five significant figures. The above sum rule applies to individual angular momentum states; if one sums over all angular momentum states, one obtains a super sum rule in which the right-hand side is a simple integer. Super sum rules for inelastic scattering from the two-electron target are also treated (in an appendix).

The interaction of a bound electron with the radiation field produced by the image charges that represent a surrounding cavity produces a shift in its orbital frequency and in its radiative decay time. We calculate the frequency shift and the change in the damping constant for a cyclotron motion at the midpoint of a lossy, cylindrical cavity. A change in the cyclotron damping constant has, in fact, been observed in the University of Washington g-2 experiments. These experiments employ electrodes with a hyperbolic shape, but our calculable model should give a useful indication of the size of the effects for them. The frequency shift can easily be so large as to have important consequences for the g-2 measurements, and this systematic effect warrants a thorough experimental investigation. As an intermediate step in this calculation, we also compute the corresponding alterations for a charged particle in the midplane between two infinite parallel conductors.

The exact representation of a two-electron wave function near the origin is the Fock expansion, i.e., a double summation over powers of R and of lnR [where R≡($r_1^2$+$r_2^2$${)}^{1/2}$] with coefficients dependent on the five remaining angular variables. Using a representation of hyperspherical harmonics, we present here the first numerical solution of the equations for the Fock coefficients. We present also a general procedure for matching a linear combination of Fock-series solutions onto a basis of adiabatic hyperspherical functions at a matching radius $R_0$. This matching procedure ensures that the proper asymptotic boundary conditions are satisfied. Exploratory numerical results are presented for ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ wave functions of He and ${\mathrm{H}}^{\mathrm{-}}$ in which four Fock-series solutions are matched onto the lowest ${}_{}{}^1{}_{}{}^{}\mathrm{S}$ adiabatic hyperspherical wave function at a matching radius near the first antinode in the adiabatic wave function.

We use Feshbach’s theory of resonances to demonstrate that bound states in the continuum (BIC’s) can occur due to the interference of resonances belonging to different channels. If two resonances pass each other as a function of a continuous parameter, then interference causes an avoided crossing of the resonance positions and for a given value of the continuous parameter one resonance has exactly vanishing width and hence becomes a BIC. The condition for a BIC relates the positions of the noninterfering resonances with the coupling matrix elements between the various channels. In the neighborhood of the BIC point one resonance remains anomalously narrow for a finite range of values of the separation of the noninterfering resonances. Whether or not two resonances interfere is not directly related to whether or not they overlap. All these results, including the occurrence of exactly bound states in the continuum, are not consequences of approximations inherent in Feshbach’s theory but are general features of a coupled-channel Schrödinger equation with only one open channel. We illustrate the results in a simple but realistic model, where all matrix elements involved can be calculated analytically. We also discuss the case of coupled Coulombic channels where BIC’s are caused by perturbations interfering with a Rydberg series of autoionizing resonances. Below the continuum threshold the analogy to a BIC is an infinitely narrow perturbation of the bound-state spectrum. Near such an infinitely narrow perturbation we may observe approximate level crossings.

We present an exactly solvable effective potential that reproduces atomic spectra in the limit of exact quantum-defect theory, i.e., the limit in which, for a fixed l, the principal quantum number is modified by a constant: $n^{\mathrm{*}}$=n-δ(l). Transition probabilities for alkali atoms are calculated using the analytical wave functions obtained and agree well with accepted values. This allows us to make phenomenological predictions for certain unknown transition probabilities. Our analytical wave functions might serve as useful trial wave functions for detailed calculations.

The use of complex coordinates allows the possibility of treating resonances of many-electron systems based on a complex eigenvalue Schrödinger equation. A many-body analysis of this equation has led to the establishment of convenient and systematic procedures for including electron correlation and for computing partial and total widths, without and with interchannel coupling, in multichannel autoionization. This is done by computing the asymptotic-pair (AP) correlation functions from an independent asymptotic-pair approximation (IAPA) and then mixing them (MAP, i.e., the mixing of asymptotic pairs) from a diagonalization of the total non-Hermitian Hamiltonian matrix. Nonorthonormality complications are resolved by straightforward computation. We report results for ${\mathrm{Ne}}^{+}$ 1s2$s^2$2$p^6$ ${}^2$S, which decays into five one-electron channels: 1s-2$p^2$ ${}^1$D, 1s-2$p^2$ ${}^1$S, 1s-2s2p ${}^3$$P^o$, 1s-2s2p ${}^1$$P^o$, and 1s-2$s^2$ ${}^1$S. The corresponding partial-width values from MAP are (in ${10}^{\mathrm{-}2}$ a.u.): 0.565, 0.040, 0.032, 0.165, and 0.043. The proximity of these results to the available experimental information as well as that to previous results from many-body calculations with real coordinates offers a practical verification of this theory.

With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schrödinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schrödinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Zα, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.

We report the first measurements on a dc superconducting quantum interference device (dc-SQUID) coupled to a gravitational wave detector cooled down to liquid-helium temperature. The overall system is composed of a 2.3-ton aluminum bar and a resonant capacitive transducer coupled to the dc-SQUID amplifier by means of a high-ratio superconducting transformer. The SQUID used in this experiment is a planar device with a multiple-loop geometry having both low noise and good coupling with the external world. The electromechanical equivalent circuit of the system, which is essentially a set of three coupled oscillators, is analyzed in detail in its normal modes and an expression for the effective noise temperature for each mode is given. Measurements are reported on the SQUID alone, on the SQUID plus the high-ratio transformer, and on the SQUID coupled to the antenna in parallel with a field-effect transistor preamplifier.

This article is concerned with the construction of a quantum-mechanical Hamiltonian describing a computer. This Hamiltonian generates a dynamical evolution which mimics a sequence of elementary logical steps. This can be achieved if each logical step is locally reversible (global reversibility is insufficient). Computational errors due to noise can be corrected by means of redundancy. In particular, reversible error-correcting codes can be embedded in the Hamiltonian itself. An estimate is given for the minimum amount of entropy which must be dissipated at a given noise level and tolerated error rate.

The analytic properties of solutions of the relativistic Thomas-Fermi equation which tend to zero at infinity are first examined, the neutral-atom solution being a member of this class. A new length is shown to enter the theory, proportional to the square root of the fine-structure constant. This information is used to develop a perturbation expansion around the neutral-atom solution, corresponding to positive atomic ions with finite but large radii. The limiting law relating ionic radius to the degree of ionization is thereby displayed in functional form, and solved explicitly to lowest order in the fine-structure constant. To embrace this knowledge of heavy positive ions, as well as results from the one-electron Dirac equation, a proposal is then advanced as to the analytic form of the relativistic total energy E(Z,N) of an atomic ion with nuclear charge Ze and total number, N, of electrons. The fact that, for N>1, the nucleus is known only to bind Z+n electrons, where n is 1 or 2, indicates nonanalyticity in the complex Z plane, represented by a circle of radius Z∼N. Such nonanalyticity is also a property of the nonrelativistic energy derived from the many-electron Schrödinger equation. The relativistic theory, however, must also embody a second type of nonanalyticity associated with the known property for N=1 that the Dirac equation predicts electron-positron pair production when the electronic binding energy becomes equal to twice the electron rest mass energy. This corresponds to a second circle of nonanalyticity in E(Z,N), and hence to a Taylor-Laurent expansion of this quantity in the atomic number Z. The relation of this expansion to the Layzer-Bahcall series is finally discussed.

The finite-element method has been used to obtain numerical solutions to the Schrödinger equation for the ground state of the helium atom. In contrast to the globally defined trial functions of the standard variational approach, the finite-element algorithm employs locally defined interpolation functions to approximate the unknown wave function. The calculation reported herein used a three-dimensional grid containing nine nodal points along the radial coordinates of the two electrons and four nodal points along the direction corresponding to the cosine of the interelectronic angle. This produced an energy of -2.9032 a.u., which lies 0.017% above the Frankowski-Pekeris value. The values of 〈$r^n$〉, for n=-2,-1, 1, and 2, are closer to those of Frankowski and Pekeris than from all of the variational calculations with the exception of the calculation performed by Weiss, whose energy and 〈$r^n$〉 values are comparable to those of the finite-element computation.

Cross sections for the electron capture by 82- , 140- , and 200-MeV/amu ${\mathrm{Xe}}^{54+}$, ${\mathrm{Xe}}^{53+}$, and ${\mathrm{Xe}}^{52+}$ ions incident on thin solid targets from Be to Au were measured. The measurements are compared with calculations of radiative and nonradiative capture. The nonradiative capture calculations are based on the relativistic eikonal approximation which includes transitions from K, L, and M shells of the target to nearly all projectile shells. In high-Z targets, nonradiative capture into excited states of the projectile is dominant, as predicted by the eikonal calculations and confirmed by comparing measured cross sections for bare ${\mathrm{Xe}}^{54+}$ and ${\mathrm{Xe}}^{52+}$ (1$s^2$) projectiles. A simple formula for estimating nonradiative-electron-capture cross sections for relativistic-ion accelerator design and other applications is derived. Double and triple electron capture are also observed in the low-velocity, high-Z collisions.

Projectile 1s ionization cross sections are reported for 82- , 140- , and 200-MeV/amu Xe projectiles incident on a variety of thin solid targets between Be and Au. The cross sections were calculated with the plane-wave Born approximation. Possible relativistic wave-function, target-screening, and transverse-excitation effects are discussed. Comparisons of the data with the perturbed-stationary-state theory of Basbas et al. and the Glauber approximation are made. Scaled, interpolated Xe+Xe ionization cross sections agree well with measured p+H ionization cross sections.

Total-electron-capture cross sections are reported for fully stripped and H-like ions of C, N, O, F, and Ne colliding with atomic and molecular hydrogen targets in the energy range 0.18–8.5 keV/amu. At energies above 3 keV/amu the cross sections are approximately energy independent with magnitudes in the range (4–7)×${10}^{\mathrm{-}15}$ ${\mathrm{cm}}^2$, and scale smoothly with projectile charge. Below 3 keV/amu the H-target cross sections for projectiles with even charge are smaller than those for projectiles of odd charge. For the ${\mathrm{H}}_2$-target data, the opposite trend prevails. For those systems where calculations exist, molecular orbital and atomic orbital close-coupling calculations are within 20% of the experimental results except at the lowest energies investigated.

Proton and helium-ion impact on atomic lithium is studied for impact energies ranging from 15 to 200 keV. By observing the final lithium charge states, cross sections for pure single electron capture from lithium as well as for single capture plus ionization are presented. Cross sections for double electron capture by ${\mathrm{He}}^{2+}$ projectiles are also measured and the decrease in the double-capture cross section below 20 keV/amu is confirmed. Cross sections for ionization of lithium by these same projectiles are also presented and compared with theoretical calculations.

Our calculations show that an external dc electric field can enhance by many orders of magnitude the depolarization cross section of highly excited atoms by charged particles. The enhancement is due to the fact that the electric field extends and shifts the electronic charge distribution along its direction, thus effectively creating a giant electric dipole in the atom.