In the bulk limit (particle number $N\to \infty$, volume $\Omega \to \infty$ with $\frac{N}{\Omega }≔\mathrm{const}$), the one-body density operator $n\left(t\right)\mathrm{}$ describing interacting distinguishable particles satisfied a closed evolution equation. This equation contains a set of terms which explicitly depend on the initial -correlation operators $\chi$, in contrast with a typical kinetic equation such as Boltzmann's equation. The nature of these terms is investigated. It is explicitly shown that the initial pair correlation ceases to influence the evolution of $n\left(t\right)\mathrm{}$ after a time of the order of a collision duration for a weakly coupled low-density gas.

The effects of a transverse mode, degeneracy, overlapping transitions, background losses, and transverse population variations are included in an analysis of light-pulse amplification in a system with an inhomogeneously broadened gain profile. Over a wide range of resonant gain and background loss, the steady-state pulse energy is proportional to the inverse cube of the pulse width, in accord with the observations of Frova et al.

Nuclear-magnetic-resonance techniques have been used to study vacancy diffusion and the exchange interaction of ${\mathrm{He}}^3$ in solid ${\mathrm{He}}^3$-${\mathrm{He}}^4$ mixtures between 0.3 and 2.0 °K for molar volumes between 20 and 22 cc/mole. The concentrations studies are 32.1, 7.78, and 1.94% ${\mathrm{He}}^3$ in ${\mathrm{He}}^4$. There is good agreement between the diffusion activation energies determined from the $T_2$ measurements and those obtained by measuring the diffusion constant directly using the field-gradient technique. These activation energies are consistently lower for mixtures than for pure ${\mathrm{He}}^3$. The $T_1$ data in the Zeeman-exchange plateau region indicate that the exchange interaction is independent of concentration. The $T_2$ data in the exchange-narrowed region are not in agreement with the theoretical result obtained by allowing the moments of the line shape to become concentration dependent. The experimental values of $T_2$ are much lower than the predicted values. This deviation is qualitatively explained by postulating the existence of two spin species: those that strongly experience the effects of exchange and those that do not. A small fraction of the isolated spins can then dominate the $T_2$ relaxation process.

We consider the radiation from a system of $N$ identical two-level atoms coupled to a continuum of quantized em modes, and possibly, to an external driving field near resonance. The atoms can be distributed over a region large in comparison to the resonant wavelength, but smaller than the spontaneous pulse length. Radiation rates and correlation functions are expressed in terms of expectation values of time-dependent atomic operators, which are shown to satisfy coupled first-order differential equations involving similar atomic operators and the initial radiation operators. The corresponding equations for the expectation values simplify considerably if no driving field is present. Similar results are derived for a model in which each atom is replaced by a harmonic oscillator.

Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms $A_1$, $A_2$, whose separation $r_{21}$ can be comparable to the wavelength $\lambda$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\pi$ pulse, and (c) only $A_1$ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators $O_1$, $O_2$. The atoms exhibit superradiant behavior, whereas $O_1$, $O_2$ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\theta ,t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\to \infty$ is the same as in the independent-atom case $r_{21}\gg \lambda$. For the oscillators, the lobes of $\mathcal{R}(\theta ,t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\Omega }_{12}$ corresponding to the shifts $\pm {\Omega }_{12}$ of the triplet and singlet states due to the $A_1-A_2$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that $A_2$ and $A_2$ radiate simultaneously around the frequencies $\epsilon \pm {\Omega }_{12}$, where $\epsilon$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.

The time evolution and its implications with regard to the approach to equilibrium of a single-particle system in a suddenly expanded box with perfectly reflecting walls is investigated using density-matrix techniques. It is shown that the time-dependent quantities exhibit a periodic behavior which is independent of the initial density matrix as long as the dimensions of the system are finite. The period is found to be $\frac{2\pi \hslash }{E_1}$ where $E_1$ is the ground-state energy of the expanded system. The continuum approximation is employed to determine the explicit time dependence of the particle density of a system initially in thermal equilibrium. It is found that the density for an expansion into a finite volume decays essentially as $\frac{1}{t}$ with small oscillations superposed on this behavior for times ranging from shortly after expansion up to one-fourth the characteristic period.

A study is made of the temporal evolution and its implications in regards to the approach to equilibrium of a system of noninteracting particles that has undergone a sudden expansion. It is assumed that the system is initially in thermal equilibrium. It is shown that the particle density approaches an equilibrium density essentially as $\frac{1}{t}$ for times shortly after expansion up to one-half the characteristic period. This period is the same as for a single-particle system. An expression is derived for the particle density that is valid for Bose, Fermi, and Boltzmann particles. The density for Bose and Fermi particles is compared with that found for Boltzmann particles.

Extending the earlier study by Goodstein, McCormick, and Dash, we have measured the heat capacities of ${\mathrm{He}}^3$ and ${\mathrm{He}}^4$ films absorbed in Ar-plated copper, at coverages $x$ ranging from 0.1 to 0.8 monolayer from 0.5 to 4.2 °K. The broad features of the results include the following: $C\left(T\right)\mathrm{}$ for ${\mathrm{He}}^4$ resembles the temperature dependence of two-dimensional Debye solids, but with characteristic temperatures ${\Theta }_D$ which decrease as $T$ falls; ${\Theta }_D$ at high coverage agrees with the value 28 °K obtained by Goodstein et al.; ${\Theta }_D$ decreases at lower $x$, but is surprisingly large (16 °K) at $x=0.1\mathrm{}$; ${\mathrm{He}}^3$ heat capacities at high coverage are quite similar to ${\mathrm{He}}^4$ films at the same areal density; at intermediate and lower coverages, $C(x,T)\mathrm{}$ of ${\mathrm{He}}^3$ is significantly different from that of ${\mathrm{He}}^4$ and displays a small peak or offset at $T\simeq 2$ °K. The data are compared with several microscopic models: localized adsorption, two-dimensional gases, noninteracting particles in a two-dimensional tunneling band, and two-dimensional solids. Each is shown to be inadequate to account for the observed variations with $T$, $x$, and isotopic mass. The data are then compared with the behavior of a two-phase film, and this is found to agree with the data on ${\mathrm{He}}^4$ over a substantial range of $x$, but it fails at low $x$. We are forced, by general arguments, to invoke surface inhomogeneity to resolve the paradox, and we examine an ad hoc model (due to Peierls) of a substrate consisting of two distinct regions, on which the helium is clustered into dense monolayer patches. The two-patch model is quite successful for ${\mathrm{He}}^4$, yielding two-dimensional characteristic temperatures ${\Theta }_1=16\mathrm{}$ °K, ${\Theta }_2=28\mathrm{}$ °K for the two species, the ${\Theta }_D$ values and the areas belonging to each fraction of the substrate being practically independent of $T$. Agreement with the two-patch model implies that much of the substrate is substantially bare of adatoms, and hence that adatoms are strongly bound in the dense surface phases. The latent heat for two-dimensional vaporization is estimated to be at least 15°. This value is about 8° greater than the heat of vaporization of bulk liquid ${\mathrm{He}}^4$, implying a large enhancement due to interactions with the substrate. We propose a possible mechanism for the enhancement; namely, a local depression of the surface by the adatoms: a "mattress effect." The two-patch model is successful in describing $C\left(T\right)\mathrm{}$ for ${\mathrm{He}}^3$ at high coverage, but $C\left(T\right)\mathrm{}$ at low coverage and the $x$ dependence over the entire range shows that ${\mathrm{He}}^3$ is significantly different from ${\mathrm{He}}^4$, but we have no explanation for the difference.

The experimental heat capacities of physically adsorbed films are increased by contributions due to desorption to the vapor phase, and these contributions are usually treated as an experimental nuisance. If the desorption contributions can be isolated from the total heat capacity, the measurements can lead to a model-independent determination of the vapor pressure and heats of adsorption of the film. The theory is applied to recent results on ${\mathrm{He}}^3$ monolayers. The vapor pressures and heats of adsorption deduced from desorption, together with the measured heat capacities, are used to estimate the absolute entropy of ${\mathrm{He}}^3$ submonolayers at 0.5 °K.

Polarization properties of interstellar OH and ${\mathrm{H}}_2$O masers are related to the nonlinear properties of the medium, including that of the weak magnetoplasma. Because the Raman-type cross-saturation of oppositely polarized modes is negligible for this Gaussian-type broad-band signal, contrary to laboratory cases, the preference for circular polarization in the brightest OH sources is attributed to another mechanism: parametric down-conversion of one Zeeman-split microwave mode to another and to an electron cyclotron wave.

In this paper we develop a microscopic theory of liquid helium-four at absolute zero. Our method of attack is a free adaptation of the Bohm-Pines technique for describing the degenerate electron gas, and is an improvement in several ways on the collective treatment of liquid helium carried through by Bohm and Salt using the same technique. In consequence of our approach, we are able to treat the collective motions rather accurately and to give a microscopic derivation of the Feynman theory of liquid helium. We also show precisely how the latter is related to quantum hydrodynamics and how the Feynman-Cohen state vector for the excited states is the natural and rigorous outcome of relaxing a little the condition that density fluctuations in the liquid do not affect its kinetic energy. The ground-state wave function is computed and shown to be, to a very good approximation, of the same form as that postulated by Chester and his collaborators. This implies strongly that Bose condensation exists in liquid helium and that it is closely related to short-range correlations in the system. Lastly, we separate the microscopic Hamiltonian operator into truly independent collective and single-particle parts and verify that the coupling between the phonon modes and the short-range correlations can induce a velocity-dependent interaction between helium atoms. This new interaction is under certain conditions an attractive one and may play a role in determining the anomalous thermal properties of the liquid.

A classical analysis of time correlations in simple fluids based on the generalized Langevin equation is presented. Formulas for the current-current correlations are developed explicitly in a region of frequencies ($\omega \sim {10}^{13}$ ${\sec }^{-1\mathrm{}}$) and wave numbers ($k\sim {10}^8$ ${\mathrm{cm}}^{-1\mathrm{}}$) which are explored in typical slow-neutron-scattering measurements. Where applicable, comparisons are made with the results of the numerical calculations of Rahman in argonlike liquids, and good agreement is generally found. The analysis is based on a hydrodynamic description of fluids involving frequency- and wavelength-dependent transport parameters. The frequency and wavelength dependence of shear and longitudinal viscosities are given explicitly for argon-like liquids.

The memory functions for the velocity, angular-momentum, and dipolar autocorrelation functions from a series of molecular-dynamics studies of liquid carbon monoxide are examined. The velocity and angular-momentum memory functions decay initially almost to zero in a Gaussian fashion. However, their long-time behavior has a much slower time dependence. The dipolar memory function from a simulation using a strong noncentral potential is approximately this system's angular-momentum autocorrelation function. Approximate velocity and angular-momentum correlation functions are generated from approximate memory functions and the results are compared to experiment. Gaussian memories based on the second and fourth moments of the corresponding autocorrelation functions give the best agreement with experiment. However, none of the approximate memories examined adequately represents the long-time behavior of the experimental memories. The static atomic radial distribution functions are given and are shown to depend upon the strength of the orientational parts of the pair potential used in the dynamics calculations. The non-Gaussian characteristics of the Van Hove self-correlation functions are examined and shown to depend on the potential and number of particles used in the dynamics calculations. The intermediate scattering function and its memory are also examined.

An expression is obtained for the quantum-mechanical second virial coefficient in the form of an inverse Laplace transform of the logarithmic derivative of the Jost function. This form is useful for the calculation of the direct part of the virial coefficient at high temperatures in cases where the Wigner-Kirkwood expansion breaks down. Explicit calculations are presented for hard spheres, the square-well potential, and the square-well potential with a hard core.

We use calculated Auger and radiative transition rates to compute $K$-shell fluorescence yields for $18\le Z\le 54$. The calculated fluorescence yields are generally higher than the most recent measurements (by as much as 8%). In addition we compare the calculated Auger rates with measured $\mathrm{KLL}$ and $\mathrm{KLM}$ intensities, the calculated $\mathrm{KLL}$ total yield with other calculations, and the calculated $\frac{\mathrm{KLX}}{\mathrm{KLL}}$ and $\frac{\mathrm{KXY}}{\mathrm{KLL}}$ ratios with measurements.

The ground state of a charged-boson system is studied throughout the entire density range in a variational formalism based on the correlated wave functions. The expectation value of the Hamiltonian operator $H$ is evaluated in the hypernetted-chain approximation using a simple trial radial distribution function which contains three variational parameters. The resulting ground-state energies are found to be higher than the exact values by 0.05 and 7.5% at high- and low-density limits, respectively. At the intermediate densities ($0.01{\leqq }_{\mathrm{rs}}\leqq 100$), the ground-state energies computed numerically lie slightly below the known results. It is conjectured, but not proved, that the extremum values of $〈H〉$ obtained in the present analysis are upper bounds to the exact ground-state eigenvalues.

Relative transition probabilities of all 30 lines belonging to the prominent $3s-3p$ transition array of Ne I have been measured in emission with a wall-stabilized arc operating in an argonneon mixture at atmospheric pressure. The data have been normalized to an absolute scale provided by recent lifetime measurements of the $3p$ levels. Our results, as well as the data from other emission measurements, have been subjected to extensive comparisons and tests, including a check for fulfillment of the $J$-file sum rule. It is found that our data exhibit a much better consistency than any other set of individual values. On the basis of this analysis as well as our uncertainty estimates, we estimate that the accuracy of our individual transition probabilities for this transition array, except for a few weak lines, is of the order of 10% on an absolute basis.

Energy and angular dependences of the elastic and inelastic scattering of electrons from ${\mathrm{H}}_2$ have been measured in the energy range 10 to 16 eV and for scattering angles ranging from 10° to 120°. Three resonance series have been investigated starting at 11.30, 11.50, and 13.63 eV, respectively. The last two series decay preferentially into one single electronic channel, while the first resonance (resonance I) decays into all energetically possible exit channels, i.e., $X{}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$, $b{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $B{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$. Throughout, stress is laid on the importance of absolute cross sections and reported values, and excitation functions have been presented in absolute units. The cross sections for the elastic channel and some inelastic channels ($X{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, $v=1\mathrm{} \mathrm{and} 2$, and the dissociative continuum $b{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$) are composed of resonant and non-resonant scattering and show pronounced and rapidly changing interference structures. Relative branching ratios for resonance series I in terms of absolute total cross sections have been listed for those channels where practically no direct scattering amplitude is present. Summarizing those results, one can state that the probabilities (branching ratios) for the decay of resonance series I into the three electronic states $X{}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$, $b{}_{}{}^3{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$, and $B{}_{}{}^1{}_{}{}^{}\Sigma _u^{+}{}_{}{}^{}$ of the ${\mathrm{H}}_2$ molecule summed over all vibrational states of the exit channel are of the same order of magnitude. From the observed angular dependence of the scattered electrons, the configuration of the resonances has been determined and compared with existing theoretical predictions.

The radial Schrödinger equation for a modified Coulomb field is separated into two first-order equations for a "phase" function and an "amplitude" function which vary only in the non-Coulomb region $r<\mathrm{}{r}_0$. At $r=r_0$, these functions are related to the mixing parameter and normalization constant of quantum-defect theory. Integrals over the range [$0,r_0$] give the scattering phase shift and the wave function's amplitude near the nucleus. Phase shifts for negative energies serve to calculate the discrete eigenvalues of an atom or positive ion. A numerical procedure for this approach is outlined.