The solution of a retarded Fokker-Planck equation is obtained by an eigenfunction expansion. Exact results are obtained for the classical case, where there is no retardation, and for the case where the memory kernel is given by a hydrodynamical model and decays at long times as $t^{\frac{-3\mathrm{}}{2}}$. In the latter case only the asymptotic result had been previously known.

Stark broadening parameters of the more prominent visible germanium lines are measured in a 1-eV shock-tube plasma. Estimated accuracies for eight observed Ge II and two Ge I lines are (10-30)%. Ionic results are compared with semiclassical predictions. Agreement with theory is excellent for two of the three observed multiplets, but unaccountably poor for the third. Moreover, widths of the two lines in this third multiplet ($5s {}_{}{}^2{}_{}{}^{}S-5p {}_{}{}^2{}_{}{}^{}P$) are grossly different from one another. Possible sources for this anomaly are discussed.

Gravitationally induced, anomalous density gradients in a fluid near the critical point are shown to have significant effects on light-scattering measurements of the decay rate of density fluctuations along the critical isochore. We have calculated the height dependence of the decay rate for xenon and carbon dioxide using a parametric equation of state and both the mode-mode coupling and the decoupled-mode theories of critical-point dynamics, and have found that the minimum in the decay rate, which does not always occur at the height of the critical isochore, can be several percent lower than the value at the critical isochore. In addition, the density gradient across the diameter of the incident light beam is shown to give rise to a measured decay rate significantly different from the value at the center of the beam, particularly at small scattering angles, under some typical experimental conditions.

The concept of scaling dimensions is important in the physics of large systems, in particular, in the statistical mechanics of critical phenomena. The main purpose of this article is to explain and formulate this important concept in a more transparent and precise fashion. Discussion is in the framework of the $n$-component classical spin model and the renormalization group. It is emphasized that not every quantity, but only special ones, called scaling variables, have well-defined scaling dimensions. In general these variables can be derived by making use of certain parameters which are the scaling fields of Wegner. The dimensions are simply related to the exponents associated with the renormalization group. We hope to extract a fairly concrete picture by a general formulation followed by explicit determination of the scaling variables in the large-$n$ limit. Dimensions are obtained to $O\left(\frac{1}{n}\right)$.

The Bogoliubov-Zubarev (BZ) formulation of the superfluid Bose liquid in terms of density as a collective variable leads to a non-Hermitian Hamiltonian ($H_{\mathrm{BZ}}$) for describing the elementary excitations. An appropriate mathematical framework for dealing with non-Hermitian operators is here employed for the first time to develop schemes for studying the elementary excitations of the Bose liquid. The energies of the ground and the first excited states associated with $H_{\mathrm{BZ}}$ in "perturbation" theory are derived. A consistent scattering theory appropriate to $H_{\mathrm{BZ}}$ is also given from which the two-roton scattering amplitude in the leading order of "perturbation" theory is explicitly deduced. A finite-temperature Green's-function theory is also presented. These results are shown to be equivalent to those based on the Sunakawa Hamiltonian. Results for the energy spectrum are also shown to be equivalent with those evaluated by means of a variation-perturbation procedure based on the method of correlated basis functions in the uniform limit.

The Marcinkiewicz theorem states that the characteristic function of a probability distribution function cannot be an exponential of a polynomial of degree larger than 2. This theorem is generalized in the present paper to (i) probability distribution functions of many variables and to (ii) probability distribution functionals when the stochastic variables are both commuting (Bose) and anticommuting (Fermi). The consequences of this theorem to certain approximation schemes in many-particle physics, involving truncation of hierarchical equations, are pointed out. These follow when one observes that the hierarchical equations such as those of many-particle Green's functions can be generated from a single equation for a Green's functional whose structure is that of a characteristic functional of a probability distribution functional. The theorem implies that this would exhibit a nonpositive behavior when certain truncation schemes are employed. Specific examples illustrating our results are drawn from the theory of an electron gas, turbulence theory, and quantum optics.

An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.

The laser heating of plasma electrons in a uniform static magnetic field via the inverse bremsstrahlung process is considered. A kinetic equation is derived, and the change in kinetic energy of the electrons is calculated. For laser radiation propagating parallel to the magnetic field and incident on cold electrons, it is found that multiphoton processes are dominant when the laser frequency is near the electron cyclotron frequency. The multiphoton absorption coefficient is found to decrease as the laser frequency approaches the electron cyclotron frequency. The presence of the magnetic field reduces the intensity threshold above which multiphoton absorption occurs.

A number of $n$-level systems behaving cooperatively is described by a boson-type second-quantization formalism which has both computational and conceptual utility, and leads naturally to a classical analog. The theory is compared with that for two-level systems and illustrated by the consideration of coupling to general-type fields. Spontaneous emission at all transition frequencies is examined, with both quantum-mechanical and classical results explicitly exhibited.

A two-step nonlinear optical interaction consisting of (i) a spontaneous frequency down-conversion process coupled with (ii) a frequency up-conversion process is treated quantum mechanically in the experimentally observed case of simultaneous collinear phase matching of both processes. The temporal behavior of the output signals is found to depend on which process, (i) or (ii), predominates. The nature of the photon statistics for each of the output modes is found to be "super-Poisson," regardless of which process, (i) or (ii), is dominant, and the zero-point fluctuations characteristic of parametric amplification are exhibited.

A simple model for an interacting liquid of particles lacking an axis of rotational symmetry is proposed. The four order parameters necessary to describe an ordered phase are identified. An ensemble of such particles is described by a mean field theory. A phase diagram showing both uniaxial and biaxial phases results. The model predicts a phase diagram similar to that of the phenomenological model of Alben.

This paper is a complete report on both the experimental and theoretical aspects of the recently discovered self-phase modulation in laser-breakdown plasmas. Mainly responsible for these effects is the sudden index change which accompanies the ionization of the gaseous medium. Two theoretical models are introduced for the phase and amplitude modulation induced by the plasma. These effects are applied to the problem of short-optical-pulse generation, with emphasis on the prospects for producing a pulse consisting of only a few optical cycles. The techniques described here have the unique advantage that the generation mechanism is linear, resulting in particularly clean, reproducible and predictable optical transients. Methods are suggested for raising the plasma nucleation intensity by further cleaning up the gas. It is shown that index dispersion of the optical components may place a lower limit on pulse duration.

It is pointed out that the cubic term in the phonon spectrum of superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ cannot be rigorously determined from specific-heat measurements.

It is argued that superradiant processes are those in which the radiative response of an atom to an excitation is appreciably influenced by interactions with its neighbors, rather than processes in which the output intensity is proportional to the square of the number $N$ of atoms. On the basis of this criterion, optical free induction is a limiting response of an inhomogeneously broadened system in which cooperative effects become very small, but in which there is a strict $N^2$ dependence. In such a system, superradiance is manifest by a departure of the output from an $N^2$ dependence. Expressions are derived for the free induction response of an atomic system to an arbitrary excitation pulse, and are reduced to simple analytic forms in certain special cases. The free induction and superradiant responses are compared under conditions in which the inhomogeneous lifetime is shorter than the superradiance time $\frac{1}{N\beta }$, and an error in a previously published curve is corrected.

We comment on the work of Jodoin and Mandel concerning superradiance in an inhomogeneously broadened atomic system of small size. We argue that for any realistic geometry, when inhomogeneous broadening is small, spatial variation of the electric field plays a major role in determining superradiant behavior. When inhomogeneous broadening is large, we agree on the neglect of this spatial variation, but argue that all radiation-reaction effects are then small, and can be estimated simply.

It is generally thought that in the high-velocity limit the dominant contribution to the cross section for charge transfer (in the forward direction) comes from a particular second Born term. It is shown that for charge transfer into states of hydrogenlike atoms the particular second Born contribution which dominates in the high-velocity limit decreases as $n^{-3\mathrm{}}$ in the limit of high principal quantum number $n$.

Analytic Hartree-Fock generalized oscillator strengths for the $1 {}_{}{}^1{}_{}{}^{}S-2 {}_{}{}^1{}_{}{}^{}P$ transition in He are presented and compared with values calculated by other methods.

A previous theory for rotational excitation of molecules is modified to include the effect of the anisotropic polarization potential in addition to the quadrupole potential. This refinement results in better agreement with some new experimental data.

The simplified Kohn-Feshbach variational method is used to study the phase shifts for the $P$-wave electron-hydrogen system. Unlike earlier calculations where the curve fitting is used to calculate the width and the shift for a resonance state, we obtained these quantities by direct integration. This permits us to study the variation of $\Gamma \mathrm{}\left(E\right)\mathrm{}$ and $\Delta \mathrm{}\left(E\right)\mathrm{}$ in the resonance region and a well-defined resonance energy position can be found. In the elastic region we obtained two resonances each for triplet and singlet states. These resonance positions compare favorably with previous accurate calculations.

Generalized oscillator strengths $f\left(K\right)\mathrm{}$ have been calculated for the $\left(1\mathrm{}{s}^{2}2s^2\right){}_{}{}^1{}_{}{}^{}S\to \left(1\mathrm{}{s}^{2}2s2p\right){}_{}{}^1{}_{}{}^{}P$ transition of the Be-like ions when $Z=4-8\mathrm{}$. A "length" formula and two versions of a "velocity" formula for $f\left(K\right)\mathrm{}$ were used in each instance. For each ion, $K$- and $L$-intrashell pair correlation functions were included in the description of the ground-state wave function, and the $K$-shell pair function was also introduced into the excited state. The pair functions were determined after the general manner proposed by Sinanoğlu. For $K=0\mathrm{}$, $f\left(K\right)\mathrm{}$ is equivalent to the optical oscillator strength and our results can therefore be compared with several other theoretical values and with experiment. The excellent agreement of our $f\left(0\right)\mathrm{}$ values when compared with experiment, coupled with the high degree of consistency amongst the $f\left(K\right)\mathrm{}$ values when evaluated, in turn, from the three formulations mentioned above, suggests that our correlated results should possess considerable reliability, particularly at the lower end of the $Z$ range. The sensitivity of the $f\left(K\right)\mathrm{}$ values to the presence of electron correlation effects has been examined by repeating the calculations using Hartree-Fock descriptions for both the ground and excited states. Comparison between our correlated and noncorrelated results revealed that the presence of the pair functions was indispensable in order to achieve agreement between $f\left(K\right)\mathrm{}$ values derived from the "length" and "velocity" formulations. For Be, the $f\left(K\right)\mathrm{}$ values were used to obtain total inelastic scattering cross sections, within the first Born approximation, for both proton and electron impact. Although our allowance for electron correlation produced significant changes in the "length"-based cross sections, "velocity" values proved to be considerably less sensitive over a large energy range for both types of projectile.