The method of the preceding paper is used to construct compact and accurate integral-transform wave functions for the four lowest excited states of the helium-like ions. Convergence studies of the energy and various expectation values are presented for the $2{}_{}{}^1{}_{}{}^{}S$, and $2{}_{}{}^3{}_{}{}^{}S$, states of the helium atom. Parallelotope parameters and energies are given for the $2{}_{}{}^1{}_{}{}^{}S$, $2{}_{}{}^3{}_{}{}^{}S$, $2{}_{}{}^1{}_{}{}^{}P$, and $2{}_{}{}^3{}_{}{}^{}P$ states of the ions from He through ${\mathrm{Mg}}^{10+}$. A brief discussion of the prospects for extending the method to many-electron systems is included.

The angular correlation between $\alpha$ particles and electrons emitted from the $L$ atomic shell during ${}_{}{}^{210}{}_{}{}^{}\mathrm{Po}$ $\alpha$ decay has been measured over an angular range from 0° to 90° for electrons in the energy range from 5 to 20 keV. The angular distribution of the $\alpha$ particles in the forward hemisphere with respect to the direction of the electrons was found to be almost isotropic with no strong forward peaking. Implications of this result on the hypothesis of electron pickup by $\alpha$ particles are discussed.

A two-configuration interaction model is developed to account for the branching ratios of the $K{L}_{2,3}^n$ radiative electron rearrangement ($K{L}^n$ RER) x-ray transitions. The mixing between the quasidegenerate $2p^{-m}$ and $2s^{-2\mathrm{}}2p^{-m+2\mathrm{}}$ $L$-shell configurations is considered both for the initial ($m=n$) and final ($m=n+1\mathrm{}$) states. The branching ratio is shown to be proportional to the square of the $2p^{-n-1\mathrm{}}$ mixing coefficient in the final $2s^{-2\mathrm{}}2p^{-n+1\mathrm{}}$ states for any number of initial $2p$ vacancies. In cases where both initial- and final-state mixing occurs, the corresponding amplitudes are shown to add destructively. Branching ratios are calculated in $\mathrm{LS}$ coupling for $Z$ varying from 10 to 30 by using the multiconfiguration Hartree-Fock method. Calculated ratios are in close accord with available experimental data for Mg, Al, and Si. The validity of the two-configuration interaction model is examined, and it is found that the $K{L}^n$ RER transitions are almost entirely due to the $2s-2p$ quasidegeneracy. The $2s\to 1s$ shake down which has recently been found to be important for two-electron, one-photon $K^2-L^2$ x-ray transitions is shown to have a negligible influence on the $K{L}^n$ RER transitions. The large widths of the $K{L}^n$ RER lines are attributed to the short lifetime of the $2s^{-2\mathrm{}}2p^{-n+1\mathrm{}}$ final states which decay by $L_1^2-L_1{L}_{2,3}M$ Coster-Kronig transitions.

Extensive new data on the even-parity, $J=0\mathrm{} \mathrm{and} 2$ bound states of Ca provide an excellent opportunity for the application of multichannel quantum-defect theory (MQDT) to these spectra. A detailed description of the experimental technique of multiphoton ionization spectroscopy is given, followed by an explicit application of MQDT to the case where interacting Rydberg series go to three distinct limits. MQDT parameter sets are obtained for each of the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$, ${}_{}{}^1{}_{}{}^{}D_2^{}{}_{}{}^{}$, and ${}_{}{}^3{}_{}{}^{}D_2^{}{}_{}{}^{}$ series of Ca, permitting quantitative determination of the configuration mixing in each bound state. Several previous assignments of members of the ${}_{}{}^1{}_{}{}^{}D_2^{}{}_{}{}^{}$ series are changed by the analysis. It is found that the quantum defects of the bound energy levels of Ca are determined almost entirely by configuration mixing, with no observable effects of recoupling. This is in contrast to the situation in the rare gases, where angular momentum recoupling has the dominant effect on the quantum defects, with configuration mixing a small additional effect.

Evolution equations for the average energy of a harmonic oscillator in interaction with a bath are derived and solved for two cases: (a) when the probability evolution obeys a master equation with a true decay representing a sink-process like radiative emission occurring simultaneously with vibrational relaxation, and (b) when the Markoffian approximation in the Heisenberg-equation analysis of relaxation is not made, allowing for short-time effects.

The method of spin-exchange optical pumping has been used to measure the ratio $\frac{g_J\left({\mathrm{Rb}}^{87}\right)}{g_J\left({\mathrm{K}}^{39}\right)}$ for the alkalis in their ${}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ ground states. Zeeman hyperfine splittings of the atomic ground states were produced in a magnetic field of 43 G, and the rf transition frequencies between the states were measured. A best fit of the data was then made using the Breit-Rabi expression for the energy levels, and constants were adjusted for best fit. Possible systematic effects of pumping light, buffer pressure, etc., were considered in obtaining the final result $\frac{g_J\left({\mathrm{Rb}}^{87}\right)}{g_J\left({\mathrm{K}}^{39}\right)}=1.000\mathrm{}01845\left(7\right)\mathrm{}$. The ${\mathrm{K}}^{39}$ $g_I$ and the $g_J$ and $\frac{g_I}{g_J}$ ratios of the rubidium isotopes obtained in check runs were in good agreement with published high-precision data.

Expressions for energy-level shifts and transition amplitudes of atoms interacting with an incoming radiation field are developed in a completely quantum theoretic way and the results are compared with others. The continued-fraction expressions reproduce both those of Swain and of Gontier et al., and the numerical results for the particular case of Bloch-Siegert shifts in mercury vapor agree with Stenholm's but go further in giving the shape of the resonances. These results are also obtained by a method of direct diagonalization of the Hamiltonian of the coupled system.

A new high-temperature millimeter-wave microwave spectrometer has been constructed and used to measure a wide range of vibrational and rotational states of silicon monoxide. This work results in accurate rest frequencies for all of the intersteller SiO maser transitions that have been observed as well as accurate measurements or predictions of all transitions that are likely to be of astrophysical interest. In addition, the Dunham spectral and potential constants are calculated for the three major isotopic species. For ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$: $Y_{01}=21\mathrm{}787.453\left(11\right)\mathrm{}$ MHz, $Y_{11}=-151.026\mathrm{}\left(11\right)\mathrm{}$ MHz, $Y_{21}=70.5\mathrm{}\left(24\right)\mathrm{}$ kHz, $Y_{02}=-29.38\mathrm{}\left(13\right)\mathrm{}$ kHz, $a_0=5.390\mathrm{}\left(22\right)\mathrm{}\times {10}^5$ ${\mathrm{cm}}^{-1\mathrm{}}$, $a_1=-2.9899\mathrm{}\left(41\right)\mathrm{}$, $a_2=5.7\mathrm{}\left(7\right)\mathrm{}$, $a_3=-9.0\mathrm{}\left(5\right)\mathrm{}$. The calculated equilibrium parameters for ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ are $B_e=21\mathrm{}787.5\left(10\right)\mathrm{}$ MHz, ${\omega }_e=1252.\mathrm{}\left(3\right)\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, ${\omega }_e{x}_e=5.96\mathrm{}\left(71\right)\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, and $r_e=1.509\mathrm{}73\left(4\right)\mathrm{}$ Å.

We have made measurements of the spectrum of resonance fluorescence emitted by sodium atoms in an intense, uniform, monochromatic field. The transition used in the experiment was between the $3{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ ($F=2\mathrm{}$, $m_F=2\mathrm{}$) ground state and the $3{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ ($F^{\prime }=3\mathrm{}$, $m_F^{\prime }=3\mathrm{}$) excited state. A separate light beam, circularly polarized and resonant with this transition, was used to prepare the sodium atoms in the $F=2\mathrm{}$, $m_F=2\mathrm{}$ magnetic sublevel. This pumping scheme was used to obtain the spectrum for both on- and off-resonance excitations. The measurements are in good agreement with Mollow's calculation. In addition, we describe conditions under which we observed asymmetric spectra similar to those of earlier experiments.

Expressions have been obtained for the multipole polarizabilities and shielding factors of the hydrogen atom by application of the time-dependent perturbation scheme developed by Wiener, Askar, and Demiralp utilizing a hydrodynamic analogy to quantum mechanics. Our expressions reduce to the exact values in the zero-frequency limit, obtained by Dalgarno with conventional perturbation theory.

Pulses of laser light which undergo Raman scattering by a molecule at a resonance are observed to consist of two components: a short pulse which lasts as long as the driving pulse, and a long tail which falls off with the lifetime of the excited state. It is shown that the recent observations of Williams, Rousseau, and Dworetsky on ${\mathrm{I}}_2$ can be fitted by theory; one has to suppose that the frequency spectrum of the driving pulses is broadened to an extent equivalent to a reduction of the stated rise and fall times by a factor of 10.

With an improved microwave magnetic resonance apparatus the Zeeman transition in orthopositronium has been remeasured and hence the ground-state fine-structure interval $\Delta \nu$ in positronium has been determined with higher precision: $\Delta \nu =203.384\mathrm{}\pm 0.004$ GHz (20 ppm). The measurements were made in ${\mathrm{N}}_2$ gas and gave the value $\left(\frac{1}{\Delta \nu }\right)\frac{\partial \left(\Delta \nu \right)}{\partial D}=(-5.7\mathrm{}\pm 1.4)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ at 23°C for the coefficient of the fine-structure density shift. Relativistic binding corrections to the positronium Zeeman transition were taken into account.

The fine-structure interval $\Delta \nu$ of ground state ($n=1\mathrm{}$) positronium has been measured to high precision, yielding $\Delta \nu =203.3849\mathrm{}\left(12\right)\mathrm{}$ GHz (6 ppm). Comparison with theory is at present limited by uncalculated terms of order ${\alpha }^2\Delta \nu$. Values for the linear coefficient of the fine-structure density shift, $a\equiv \left(\frac{1}{\Delta \nu }\right)\frac{\partial \Delta \nu }{\partial D}$, have been measured for positronium in nitrogen and the noble gases: $a\left({\mathrm{N}}_2\right)=(-3.2\mathrm{}\pm 0.5)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ (0°C), $a\left(\mathrm{He}\right)=(+1.6\mathrm{}\pm 0.3)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ (0°C), $a\left(\mathrm{Ne}\right)=(-1.9\mathrm{}\pm 0.4)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ (0°C), $a\left(\mathrm{Ar}\right)=(-6.2\mathrm{}\pm 0.3)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ (0°C), $\left|a\right(\mathrm{Kr})=(-6.6\mathrm{}\pm 0.5)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$(0°C), and $a\left(\mathrm{Xe}\right)=(-7.3\mathrm{}\pm 1.7)\times {10}^{-5\mathrm{}}$ ${\mathrm{atm}}^{-1\mathrm{}}$ (0°C). The positronium density shifts differ markedly from those of hydrogen.

We consider in this paper the Doppler-broadened absorption by two-level molecules of a gas from a weak traveling electromagnetic plane wave in the presence of an oppositely traveling strong plane wave at the same frequency. Here the absorption spectrum exhibits a dip that broadens and deepens as the gas is increasingly saturated by the strong wave. This saturation effect has been employed in the recent experiments on the fine structure of atomic hydrogen performed by Hänsch, Shahin, and Schawlow (HSS). The form, linewidth, and strength of saturation dips in the fine-structure lines are studied for experimental parameters appropriate to the hydrogen experiment. In addition, saturation line-shape curves are calculated for experimental situations in which the strengths of the stronger field give rise to stronger or weaker saturation of the gas medium and varying degrees of Doppler broadening exist. The calculation of shape of the saturation dip and absorption curves for a Doppler-broadened gas of two-level atoms is made exactly; in effect, the strong field is treated as arbitrarily strong by avoiding a perturbation-theory treatment in favor of the exact solutions for a driven two-level system. The general behavior found is in accord with intuition and experimental observations by HSS. There is an optimum saturation power which gives a prominent dip and beyond which the dip suffers "power broadening."

Experimental results on the effects of finite-aperture detectors on the statistics of Gaussian light are presented. A laser illuminating a rotating diffusing screen was used to provide a source with a long coherence time, and the photoelectron statistics of a photomultiplier detector were recorded electronically. The data, which were taken with circular apertures and a linear polarizer in front of the detector, agree with calculations of factorial cumulant correction factors published by Cantrell and Fields.

A new method is given for the evaluation of higher-order perturbation corrections to energy of the nonrelativistic hydrogen atom in an external field. The choice of a suitable basis in the Hilbert space of bound states for nonrelativistic hydrogen atom is exploited in the method. The infinite series which represents the higher-order energy correction in the Rayleigh-Schrödinger perturbation theory may be expressed in such a basis in terms of finite series which are easily summed. As an illustrative example the summation of a few series, which are of importance for higher-order corrections to energy of the H atom in an external homogeneous magnetic field, is carried out.

The theory of the Forrester effect is reexamined in both classical and quantum terms. We show that temporal coherence (light beating) can arise from the interference of fields produced either by single atoms or by pairs of atoms. The singles terms, which dominate for a collisionless source with low intensity, give a light-beating signal-to-noise ratio of order unity, independent of the intensity and wavelength. This signal is also unaffected by Doppler broadening. The doubles terms, originally analyzed by Forrester, give a signal-to-noise ratio proportional to the intensity and to the square of the wavelength, and subject to Doppler broadening. Our results suggest the possibility of a new type of rf and microwave spectroscopy for excited atoms.

This paper discusses the suitability of a boson-second-quantization (BSQ) formalism for the description of cooperative atomic behavior—such as that encountered in quantum optics or laser-type phenomena. The physical significance of BSQ energy states and atomic coherent states, as well as the corresponding first quantization descriptions, are discussed. Specialization to the case in which the individual atomic systems are characterized by two levels allows comparison of the BSQ theory with other theories of cooperative behavior, not all of which can be generalized to the case of more than two levels. Passage to the classical limit, an analytical process that must be possible for macroscopic phenomena, is shown to be accomplished very simply in the BSQ formalism by conversion of the small number of collective (quantum mechanical) variables to classical random variables which are prescribed according to the quantum state of the total system. It is found that conventional semiclassical theory is a special case of the fully classical limit of the BSQ formalism, the specialization being the requirement that the system be in an atomic coherent state. The justification for the identification of coherent states with a classical description and energy states with a quantum-mechanical description, found in some of the literature on two-level systems, is questioned.

Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field $E$, and the existence of a spectral component at the beat frequency in $E^2$. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.

The density-matrix formalism is used to develop the theory of two-photon-resonant third-harmonic generation in metal vapors taking into account saturation, ionization, laser-induced Stark shifts, and variations in the wave-vector mismatch as a result of population redistribution. It is shown that for third-harmonic generation in the ultraviolet and the vacuum-ultraviolet regions, where $3\omega$ exceeds the ionization limit of the metal vapor, two-photon-resonant three-photon ionization will be the main limiting process. This will also be true for four-wave parametric processes. For efficient energy conversion, the two-photon-resonant metal vapor must be phase matchable to allow large vapor density and cell length. Otherwise, the low incident intensity limits, set by the saturation and the ionization effects, cannot be efficiently converted in only one coherence length. The theory is applied to two-photon-resonant tripling of the ruby laser in cesium vapor.