We study the thermal radiation field in finite cavities bounded by perfectly reflecting walls. Previous analytical results on the spatially averaged temporal correlation tensors for the cube-shaped blackbody are generalized for cuboidal cavities of arbitrary edge lengths including the infinite slab resonator. Computational results for very small cubes and sufficiently long times demonstrate the almost periodic character of the coherence function. The transition from cyclic to aperiodic relaxation is shown for the visibility, the phase, and the field autocorrelation. Improved results on the spectral mode density and total radiation energy are obtained as byproducts.

We develop a fully quantum-mechanical theory for the organic-dye-solution laser, obtain density-matrix equations of motion for the single-mode radiation-density operator and the matter-density operator, and solve and investigate the steady-state case. We generalize the usual Born-Markoff approximation master equation for two matter states to include four matter states, each one of which interacts with the laser radiation field. This allows us to treat exactly the organic-dye molecular triplet-state levels which participate in the laser operation in an essential way. For experimentally realizable conditions the steady-state solution contains features which are qualitatively different from nondye lasers. These effects are directly attributable to intensity-dependent triplet-state radiation absorption losses. At threshold the diagonal matrix elements of the radiation photon distribution ($R_n$) can be a decreasing function of the photon number $n$ with an inflection point rather than the usual truncated Gaussian. This necessitates redefinition of threshold. For pumping just above threshold there may be both a maximum and a minimum in $R_n$ rather than just a maximum as in usual laser theories. We also specify the effect of triplets on the narrowing of $R_n$ for pumping above threshold and the subsequent widening of $R_n$ for pumping well above threshold. Many of our results require photon-counting experiments for verification. Also, our equations are easily reducible to a semiclassical theory, where our treatment of the triplets variables is an improvement over the usual rate equations.

Photon echoes, observed recently in ruby and in ${\mathrm{NH}}_2$D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.

The Dicke Hamiltonian is linearized by expanding the field and atomic shift operators about disposable $c$- number parameters. When these parameters are chosen to be the expectation values of the corresponding operators with respect to the linearized Hamiltonian, the difference between the free energy of the original Hamiltonian and the free energy of the linearized Hamiltonian is minimized. This difference (per particle) vanishes in the thermodynamic limit; so the equilibrium statistical mechanics of the system is described by the linearized Hamiltonian. The disposable parameters are order parameters of the linearized Hamiltonian. The field and atom order parameters obey a system of coupled nonlinear equations characteristic of meanfield theories. These equations determine the critical temperature and the ordered-state behavior. The thermodynamic state of each subsystem is a statistical superposition of thermal noise and a coherent state produced by the complementary subsystem acting as a classical source. The classical driving terms are the order parameters. The order parameter and coherent-state parameter are equal for the field subsystem but not for the atomic subsystem. The ordered state is characterized by an enhanced condensation of the atomic subsystem into the state of maximum cooperation number $r$. This enhancement is attributed entirely to the Starksplitting of the atomic energy levels due to the classical driving field. A different but equivalent system of coupled nonlinear order-parameter equations is derived from the order-parameter equations of motion in thermodynamic equilibrium in the thermodynamic limit. This alternative form of the coupled self-consistent equations leads to a very simple method for locating the critical temperature and determining the ordered-state behavior of interacting systems. This method is illustrated by determining the gap equations for three model Hamiltonians. These describe (a) interaction of spin- $j$ systems with a single mode of the radiation field, (b) interaction of two-level atoms with a finite number of modes of the radiation field, and (c) interaction of two-level atoms with one or two modes of the radiation field through double photon absorption and emission processes.

Since for echo formation the off-diagonal density matrix element ${\sigma }_{\mathrm{ab}}$ evolves from ${\sigma }_{\mathrm{ba}}={\sigma }_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\varphi -\omega t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\pi }{2}$ pulse is $R_1$, the radius of curvature of the subsequent $\pi$ pulses $R_2$, $R_3$, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{R_{E1\mathrm{}}}=\frac{2}{R_2}-\frac{1}{R_1}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{R_{E2\mathrm{}}}=\frac{2}{R_3}-\frac{2}{R_2}+\frac{1}{R_1}$, and for subsequent Carr-Purcell echoes by $\frac{1}{R_{E3\mathrm{}}}=\frac{2}{R_4}-\frac{2}{R_3}+\frac{2}{R_2}-\frac{1}{R_1}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with $r^{-3\mathrm{}}$ and $r^{-6\mathrm{}}$ interactions between colliding molecules are compared with the experimental data for ${\mathrm{CH}}_3$F and with an $r^{-6\mathrm{}}$ potential with the data for ${\mathrm{SF}}_6$ and ${\mathrm{SiF}}_4$.

A stochastic parameter which appears to be related to the Kolmogorov entropy is computed for a system of $N$ particles in a line with the nearest-neighbor Lennard-Jones interaction. It is found that the parameter depends on the initial conditions, and is equal to zero or to a positive value which depends on the specific energy $u$. A limit seems to exist for the parameter at fixed $u$ when $N\to \infty$, as shown by computations from $N=10\mathrm{} \mathrm{to} 200$.

A mean-field calculation based on a simple model is reported. The numerical results indicate that only a relatively small polar interaction is required to induce a transformation from a nonpolar uniaxial smectic phase (smectic $A$) to a uniaxial smectic phase with polar layers within a reasonable temperature range. The polar order may accompany the smectic phase in all or in part of its range of existence. Arora, Taylor, and Ferguson (ATF) observed that some compounds show two phases with the smectic-$A$ morphology and suggested that the lower-temperature phase has polar order. Our results can reproduce the main features as far as the transition temperatures are concerned. The small transition entropies reported by ATF are in agreement with our results; however, there are not enough data to permit any correlation with the model parameters.

For certain interactions of a boson or fermion particle in quantum statistical mechanics any correlation function is bounded in absolute value by the corresponding free-boson correlation function (in the Euclidean region).

We consider the linear transport equation, which describes the interaction of a rarefied gas with a denser gas. We require the spatial variation of all quantities in this equation to be small over the distance of a typical mean free path, and the time variation of all quantities to be small during a typical mean free time. We also require the average velocity of the denser gas, the external electric field, and the internal sources to be small. These smallnesses are expressed analytically by means of a small parameter $\epsilon$. A solution of the transport equation, subject to the above restrictions, is constructed which is asymptotic with respect to $\epsilon$. To leading order the solution has the form $\psi \sim A_0(\overrightarrow{\mathrm{r}},t)\phi (\overrightarrow{\mathrm{r}},v,t)$, where $\phi$ is the eigenfunction of a certain operator and $A_0$ satisfies a diffusion equation containing effects due to the motion of the denser gas, inhomogeneities in the denser gas, and the external electric and magnetic fields. We discuss the physical interpretation of this equation and give expressions for the density and average velocity of the rarefied particles.

Two sets of finite-temperature relativistic fluid equations are obtained by taking moments of the Vlasov equation, using equilibrium and monoenergetic distribution functions. The closed sets of fluid-Maxwell equations are reduced to a simple set of equations under steady-state conditions, using two fluid constants of the motion derived for each set. The set obtained using the monoenergetic distribution is parametrized in cylindrical coordinates for high-current diode studies. The radial scale length for a radial equilibrium superpinch is obtained in terms of macroscopic diode parameters, and radial profiles of the pinch are obtained by solution of the one-dimensional system. It is found that high-current superpinches are characterized by a hot uniform-density core surrounded by a hollow current sheet, and that there is a limit to the current which can be propagated for a given pinch radius, given the macroscopic diode parameters. The minimum pinch radius obtainable in a diode in the steady state is obtained from the hot pinch limit, and diode scaling laws are presented, assuming Child-Langmuir or parapotential flow.

A study of the dynamics of a discrete two-dimensional system of classical particles is presented. In this model, dynamics and computations may be done exactly, by definition. The equilibrium state is investigated and the Navier-Stokes hydrodynamical equations are derived. Two hydrodynamical modes exist in the model: the sound waves and a kind of vorticity diffusion. In the Navier-Stokes equations one obtains a transport coefficient which is given by a Green-Kubo formula. The related time correlation function has been calculated in a numerical simulation up to a time of the order of 50 mean free flights. After a short time of exponential decay this time correlation behaves like $t^{-S}$, the exponent being compared to theoretical predictions.

Solutions are given for the interaction between a three-level system and two arbitrarily intense fields symmetrically detuned from the corresponding atomic resonances. The method uses a wave function and is substantially simpler than the density-matrix Bloch-vector analysis of Brewer and Hahn. It is also less general in that the level ($T_1$) lifetimes have to be taken equal to the dipole ($T_2$) lifetimes, and in that nonzero steady-state level populations cannot be obtained in the presence of this damping.

Using the beam-gas method with $\mathrm{O}_2^{}{}_{}{}^{+}$ and ${\mathrm{O}}^{+}$ beams incident on a differentially pumped gas-jet target, we have reinvestigated some visible lines of OII and have obtained the following radiative lifetimes (in nsec): $3p{}_{}{}^4{}_{}{}^{}P_{}^o{}_{}{}^{}$, 6.0 ± 0.5; $3p{}_{}{}^2{}_{}{}^{}P_{}^o{}_{}{}^{}$, 7.3 ± 0.7; $3P^{\prime }{}_{}{}^2{}_{}{}^{}F_{}^o{}_{}{}^{}$, 11.3 ± 0.8; $3d{}_{}{}^4{}_{}{}^{}F$, 5.0 ± 0.4; and $3d^{\prime }{}_{}{}^2{}_{}{}^{}G$, 5.0 ± 0.5.

We study the thermodynamic properties of two-level atoms interacting with all the modes of the electromagnetic field. We show that the presence of the $A^2$ term in the interaction Hamiltonian has two important consequences: The divergence difficulties noted previously for the all-mode problem are overcome, so that a formal thermodynamic limit exists; and the phase transition, which occurs in the single-mode case without the $A^2$ term, is absent.

On the basis of a variational scheme for a trial wave function given as an expansion of the exact solutions of the three-dimensional isotropic harmonic oscillator, we have calculated the energies for the 14 lowest energy states of a hydrogen atom in the presence of magnetic fields ranging from approximately ${10}^9$ to ${10}^{11}$ G. We also discuss the connection between the low- and high-field energy spectra based upon the noncrossing rule.

Criticisms are given of a recent theory by Whealton of ion transport for a slightly ionized rarefied gas in a strong electric field. His prediction of shocklike phenomena in ion swarm experiments is shown to be untenable, being the result of an unphysical assumption.

Berge and Skullerud's criticisms arise from an incomplete reexamination of the lowest nontrivial level of the transport theory under question. A closer examination shows their criticism to be unfounded. However, even if true, the value of their criticism is dubious since the theory would still be at worst asymptotically valid for small density gradients. This would serve to at least delineate the region of validity of the conventional diffusion analysis of drift-tube experiments.

The non-closed-shell many-electron theory (NCMET) of Sinanoğlu is applied to excited states which are not the lowest of their symmetry. Variational collapse towards lower states of the same symmetry is prevented by use of the Hylleraas-Undheim-MacDonald theorem. The procedure for calculating such states is described. Using it, variationally correct charge wave functions are obtained and for the first time for several $2s2p^n$ excited states in neutral and singly ionized atoms of carbon, nitrogen, oxygen, and fluorine. The configuration labels of these states and lower energy states of the same symmetry are found in agreement with the experimentally assigned orderings in each case. Optical oscillator strengths are calculated with these wave functions and are found in good agreement with experiment where previous calculations in the literature had shown large discrepancies.

The cross sections for the electron-impact-induced resonance transition ($2s\to 2p$) in lithium are calculated in the Glauber approximation using Franco's method of explicitly involving all the electrons of the many-electron target. Our results are always a bit smaller than the frozen Glauber results, but approach them as the energy increases. However, these differences are not very significant, indicating that the inner electrons (those in $1s$ state) are rather inert. Both of the Glauber results are found to be in good agreement above 15 eV with the recent experimental results of Leep and Gallagher.

Approximate cross sections for charge transfer between ground-state atomic hydrogen and completely stripped ions of C, N, O, Ne, Si, and Ar have been obtained in the velocity range 6 × ${10}^6$ - 7 × ${10}^7$ cm/sec. In these collisions, electron capture occurs predominantly into high-lying orbitals of the product heavy ion. The calculations are made using the Landau-Zener theory modified for application to a multistate system. The peak cross sections are found to increase by about a factor of 5 in going from ${\mathrm{C}}^{+6\mathrm{}}$ (≃ 21 ${\mathrm{\AA }}^2$) to ${\mathrm{Ar}}^{+18\mathrm{}}$ (≃ 110 ${\mathrm{\AA }}^2$) and the cross-section curves tend to "flatten out" for the heavier ions where three or four crossings contribute to the charge exchange. At the higher energies, the cross sections become roughly equal to 1.4 × ${10}^{-16\mathrm{}}$${\mathrm{Z}}^{3/2}$ ${\mathrm{cm}}^2$.