In this paper we are concerned with the propagation of a light beam through an inhomogeneous, isotropic medium with a possibly nonlinear index of refraction. The customary paraxial approximations of neglecting grad $\mathrm{div}\mathcal{E}$ and seeking a plane-polarized solution are shown to be incompatible with the exact Maxwell equations. By starting from Maxwell's equations, and scaling transverse and longitudinal distances by the beam waist $w_0$ and diffraction length $l$, respectively, an expansion procedure in powers of $\frac{w_0}{l}$ is developed. The exact equations obeyed by the zeroth-order fields are not Maxwell's equations but the customary paraxial approximation to Maxwell's equations. Equations for the first-, second-, and third-order fields are developed. The first-order field is found to be a longitudinal field. It is solved for explicitly in terms of the zeroth-order field which is transverse. Thus a precise knowledge of the meaning and accuracy of paraxial wave optics is obtained.

The Gordon conjecture that the mean-square fluctuations in correlation functions follow a local equilibrium rule, is shown to hold true for the simple case of an oscillator coupled linearly to its surroundings. It is suggested that the method of solution, which involves projection-operator techniques, may be applicable to the multitime correlation functions which are needed to describe nonlinear response.

The stimulated line of the Rayleigh wing (SLRW) was studied with circularly polarized light from a giant-pulse ruby laser focused into nitrobenzene. The dependence of the observed frequency shift at high input power on the distance of the focus from the entrance window reported previously was confirmed. At low power the shift is constant regardless of position, consistent with existing theory. For intermediate power a complex spectrum is observed. The simultaneous observation of self-focusing and the SLRW provided an impetus for the modification of the theory to include both phenomena. Experiments were performed to test the predicted absorption dependence. Experimental results indicate substantial agreement.

We propose and analyze the use of quantum-beat phenomena to test neoclassical radiation theory (NCT) and quantum electrodynamics (QED). For a beam-foil type of experiment with atoms having one upper level and two closely spaced lower levels, all coherently excited, NCT predicts the presence of quantum beats in the emitted radiation; beats are not expected in QED. QED predicts beats when many atoms are present, in agreement with recent photon-echo experiments. An experiment to test NCT and QED is suggested.

The theory of spontaneous decay is studied using both quantum electrodynamics (QED) and semiclassical theories of radiation. There are qualitative differences between the theories in the prediction of interference phenomena. In QED, systems which have been excited with pulsed laser light do not exhibit quantum interference effects associated with lower state splittings. On the other hand, semiclassical treatments of spontaneous decay do indicate the existence of interference effects not present in QED. In addition to this, differences are found between the predictions of fluorescence intensity in the presence of lower-state level crossings under continuous excitation.

The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.

The quantum theory of a driven damped harmonic oscillator is presented. By using iteration methods the time development of the density operator for the oscillator is obtained and its normally ordered generating functional is written. The field of the oscillator appears to be the superposition of three fields. The first field, which shows the damping of the oscillator, is statistically similar to the one before the interaction; the second one is a coherent field that depends on the driving field; and the third describes a thermal field. In order to evaluate two-time averages by differentiation, a two-time generating functional of the oscillator is obtained. This functional allows two-time averages to be computed from one-time averages before the interaction.

A simple extension of the Ornstein-Zernike theory of critical scattering gives rise to correlation functions which do not scale. The critical-point exponents have values $\eta =0\mathrm{}$ and $2\nu \ge \gamma$.

We present a new series of calculations in the theory of spinodal decomposition. The computational scheme is based on a simple ansatz for the two-point distribution function which leads to closure of the hierarchy of equations of motion for the high-order correlation functions. The resulting theory is accurate throughout the spinodal region of the phase diagram, including at the boundaries of this region where the spinodal mechanism is difficult to distinguish from nucleation and growth. The computational scheme is worked out in detail for parameters approximating those of the three-dimensional, kinetic, spin-exchange Ising model with nearest-neighbor interactions. Numerical agreement with recent Monte Carlo data appears to be satisfactory.

Computer "experiments" are used to determine the equilibrium molecular structural and thermodynamical quantities, collision rates, and the time dependence of linear and angular momentum autocorrelation functions. The properties of a heavy (Brownian) rough sphere are also studied. The initial time dependence of the autocorrelation functions obeys the predictions of Enskog's binary-collision model. After a few (∼12) multiples of the molecular collision time, the time dependence of the linear-momentum autocorrelation function is consistent with the predictions of the fluctuational hydrodynamical model, successfully applied earlier to smooth-sphere fluids. The same model, under the assumption that the time scales for relaxation of linear and angular momenta are well separated, does not agree with experimental results on the time dependence of the angular momentum autocorrelation function.

The quasielastic scattering of laser light from nematic MBBA is studied utilizing homodyne procedures. The results stem from a comprehensive study of the half widths and areas of the "slow mode" peaks together with their temperature and angle dependence. The data taken at small angles yield the temperature-dependent ratios of the viscosity coefficients to curvature elastic constants for twist, bend, and splay. From these ratios, utilizing data of others, the temperature dependence of the twist elastic constant $K_{22}$, and twist, splay, and bend viscosity coefficients ${\gamma }_1$, ${\eta }_s$, and ${\eta }_b$, respectively, are determined. From the data taken at large angles the values of ${\alpha }_4^{\prime }$ and ${\alpha }_v^{\prime }$ (which is found to be negative) are determined at a reduced temperature, $T_R=0.96\mathrm{}$.

The rate of energy dissipated by a harmonically bound Brownian particle also subjected to an ac external field is considered with the use of a non-Markoffian kinetic equation which results from including memory effects in the description of the particle motion. We find a general expression, Eq. (8), for the power dissipation and also give some numerical results for comparison with earlier results obtained using a Markoffian kinetic equation. The Brownian-particle velocity autocorrelation function is also briefly discussed.

The coherent scattering function is calculated for liquid rubidium at its melting point on the basis of the theory developed in paper II of this series and using the static structure factor obtained by Rahman from computer simulation with a potential determined by Price et al. It is noted that for $q$ up to 1 ${\mathrm{\AA }}^{-1\mathrm{}}$, the theory indicates the existence of propagating density waves unlike liquid argon, but the shape of $S(q, \omega )$ is found to disagree with the experimental observations. For intermediate values of $q$, the agreement between the theoretical and experimental results is not so good, but the disagreement is always less than 40%. However, for $q\ge 3$ ${\mathrm{\AA }}^{-1\mathrm{}}$, the theoretical results for $S(q, \omega )$ are quantitatively in good agreement with the recent neutron inelastic-scattering measurements on liquid rubidium.

The exact density of states for one particle in the field of an attractive $\delta$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature $T_c$. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, $E_b$. The transition temperature and the discontinuity of the specific heat at $T_c$ are also obtained as a function of $E_b$.

Systematic correction procedures to a recent noniterative calculation of the classical electron-gas radial-distribution function are presented and discussed. Numerical results are given in both tabular and graphical form.

We have observed the $K$-shell-associated noncharacteristic radiation emitted in Al-Al and Ni-Ni collisions in the range 0.1-0.75 MeV/nucleon and determined the production cross section. We find that most of the predictions of Lichten regarding the cross section are borne out by the experimental data.

The scattering of electrons by target atoms with $Q$ electrons is considered in the full eikonal approximation. An expression for the scattering amplitude is given which reduces the number of integrations from $3Q+3 \mathrm{to} 2Q$.

Autoionizing levels in cesium, rubidium, and potassium have been studied by electron impact in a crossed-beam apparatus. Comparisons are made with the binary-encounter calculations of Roy and Rai. The effects of autoionization and inner-shell ionization have been overestimated in the theory.

Perturbation and extrapolation techniques based on expansion in inverse powers of the atomic number $Z$ have recently been used to estimate term values for $S$ and $P$ states of heliumlike atoms in the intermediate range of values of $Z$. The coefficients in these expansions are determined by utilizing the accurate data currently available for values of $Z\le 10$. In order to obtain optimum results from such techniques, the data for low values of $Z$ should exhibit a smooth variation with atomic number. To meet this requirement, the paper lists smooth values for the nonrelativistic energy and the relativistic corrections for the ground state and the states $n{}_{}{}^1{}_{}{}^{}S$, $n{}_{}{}^3{}_{}{}^{}S$, $n{}_{}{}^1{}_{}{}^{}P$, and $n{}_{}{}^3{}_{}{}^{}P$, $n=2-5\mathrm{}$ for $Z=2-10\mathrm{}$. Smooth behavior with $Z$ has been ensured by quoting, for a given state, values obtained using the same type and length of expansion for the wave function. Also, the effect of rounding errors has been minimized by quoting the original values obtained in atomic units, and by listing the numbers to one or two digits more than would be justified by convergence arguments.

The yields for sulfur, scandium, iron, and copper $K$ x rays produced in collisions with oxygen and fluorine projectiles have been measured over a large range of impact velocities (${\eta }^{\frac{1}{2}}\simeq 0.2-0.8$). These yields, when scaled by the corresponding velocity-matched protoninduced yields, $\frac{\sigma \mathrm{}\left(Z\right)\mathrm{}}{Z^2\sigma \mathrm{}\left(1\right)\mathrm{}}$, all cross the unity value at ${\eta }^{\frac{1}{2}}$ above the values suggested in a previous theoretical paper but do approach increased-binding predictions with decreasing ${\eta }^{\frac{1}{2}}$.