We investigate the quantum electrodynamics of many-body systems, and discuss the circumstances under which the results are identical to the results of classical electrodynamics. This classical behavior is shown to occur when the motion of a large number of the particles is correlated. The correlation manifests itself in a form of coherence which is characterized by the many-body quantum state's being very nearly an eigenstate of the current operator. We present several explicit examples of quantum states which exhibit such coherence and show that they give the expected classical results. Nowhere do we assume $h\to 0$.

We have studied the $e$-beam-excited molecular-hydrogen laser and the effects of foreign gases, preionization, and cooling the gas. The pressure dependence up to 200 Torr of the laser output and the influence of the impurity gas are described in terms of the energy distribution of the plasma electrons produced in the gas breakdown due to the electric field induced by the fast-rising current pulse. Small additions of argon and nitrogen sharply reduce the laser output, unlike helium which has little effect. Preionization of the gas destroys the lasing action which fully recovers in a time of the order of a millisecond. High-repetition-rate three-level laser action should be possible. Magnetic neutralization of the induced electric field or destruction of inversion would both account for the loss of laser action. The optical gain is increased to more than 300% by liquid-nitrogen cooling. The gain depends on the reduction of Doppler broadening by cooling but must also be understood in terms of the thermal redistribution of the rotational populations. Arguments are given in favor of the direct electron excitation of lasing levels in contrast to excitation involving charged intermediate states.

Onefold photoelectron counting distributions are evaluated for coherent light scattered by an ensemble of particles. The ensemble contains, in general, a random number of nonidentical particles. Expressions are obtained for the probability density function and the moments of the scattering intensity. These results reduce to those of Pusey, Schaefer, and Koppel when the particles are identical. Photocount probabilities are also evaluated. The variance, coefficient of skewness, and coefficient of excess are evaluated when the probability density of the intensity contributed by each scatterer is specified by a narrow Gaussian distribution centered about the deterministic mean.

The behavior of self-diffusion in Kr and ${\mathrm{CH}}_4$ at intermediate densities is discussed in comparison with the Lennard-Jones and the hard-sphere fluid. Additional measurements of self-diffusion in Kr at low densities are reported.

The electron-ion recombination mechanisms of ${\mathrm{He}}_2^{+}$ are determined along with the rate coefficients of the important elementary processes which govern the relaxation of the helium afterglow, at room temperature. The experimental data (atomic- and molecular-ion currents to the walls, atomic and molecular metastable concentrations, electron concentration, elastic electron collision frequency, electron radiation temperature) obtained as a function of time under a wide range of experimental conditions are compared with the solutions of a system of five coupled partial differential equations which includes all the processes occurring in a helium afterglow. A unique set of rate coefficients and constants is found allowing the precise reproduction of all the experimental data obtained at seven pressures from 5 to 100 Torr. The electron energy balance and electron energy distribution function are calculated as a function of time and space. It is shown that the spatial distribution of the electron energy in our cylindrical experimental cell is not uniform and has to be taken into account, as well as the influence of the non-Maxwellian electrons. The recombination rate coefficient for ${\mathrm{He}}_2^{+}$, given under the form ${\alpha }_2=({\alpha }_{c2\mathrm{}}+k_{02}{n}_0){\left(\frac{T_e}{293°\mathrm{K}}\right)}^{-x_2}+k_{e2\mathrm{}}{n}_e{\left(\frac{T_e}{293°\mathrm{K}}\right)}^{{-y}_2}$, is found to be such that ${\alpha }_{c2\mathrm{}}<5<{10}^{-10\mathrm{}}$ ${\mathrm{cm}}^3$ / sec, $k_{02}=(5\mathrm{}\pm 1)\times {10}^{-27\mathrm{}}$ ${\mathrm{cm}}^6$ / sec, $k_{e2\mathrm{}}=(4.0\mathrm{}\pm 0.5)\times {10}^{-20\mathrm{}}$ ${\mathrm{cm}}^6$ / sec, $x_2=1\mathrm{}\pm 1$, $y_2=4.0\mathrm{}\pm 0.5$. These coefficients correspond to a collisional-radiative model for the recombination of ${\mathrm{He}}_2^{+}$ with electrons, which strongly depends on pressure, electron concentration, and electron temperature. 70% of the recombined molecular ions produce atomic metastables corresponding to a dissociation in the lower excited states of the molecule. The rate coefficients for ionizing collisions between metastables are found to be ${\beta }_{11}=(1.5\mathrm{}\pm 0.3)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$ / sec, ${\beta }_{22}=(1.5\mathrm{}\pm 0.5)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$ / sec, ${\beta }_{12}=(2.5\mathrm{}\pm 1.5)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$ / sec. The superelastic electron-metastable rate coefficients are ${\gamma }_1=(4.2\mathrm{}\pm 0.6)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$ / sec and ${\gamma }_2=(3.8\mathrm{}\pm 0.8)\times {10}^{-9\mathrm{}}$ ${\mathrm{cm}}^3$ / sec. All the rate coefficients compare very well with available theoretical data. The method used gives a complete solution of the helium afterglow at room temperature. It can be extended in pure helium to many other experimental conditions and applied to the study of afterglows in other pure gases or mixtures.

Using the thermal-lens effect, the specific heat $C_p$ and the thermal conductivity $\Lambda$ have been measured simultaneously in the isotropic phase of the liquid crystal MBBA ($p$-methoxy benzylidene $p$-butylanyline) near its isotropic-to-nematic transition. Accurate analysis of the lens dynamical behavior yields both transport coefficients provided the change of the refractive index with temperature, $\frac{\mathrm{dn}}{\mathrm{dT}}$, is known. A precise measurement of $n\left(T\right)\mathrm{}$ was carried out separately. The best fits yield the critical exponents 0.52 and 0.65 for $C_p$ and $\Lambda$, respectively, with a critical temperature $T_c$ equal to $T_p-0.21\mathrm{}\pm 0.01$°C, where $T_p$ is the clearing temperature. The present measurements give the first conclusive results on the critical behavior of $C_p$ in the isotropic phase of a liquid crystal.

A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential $V\left(x\right)\mathrm{}$. For the oscillator potential $V\left(x\right)=\frac{k{x}^2}{2}$ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V\left(x\right)=-\frac{A{x}^2}{2}+\frac{B{x}^4}{4}$, is calculated over a range of the positive constants $A$ and $B$ and compared with numerical solutions of the Schrödinger equation.

The moderately dense quantum Boltzmann gas at equilibrium is shown to be equivalent to a gas of "dressed particles," or "quasiparticles," like those of Landau's theory of a Fermi liquid. The Landau quasiparticle interaction energy $F\left(p\right)\mathrm{}$ is related to the second virial coefficient by means of an inverse Laplace transform. Expressions are obtained for $F\left(p\right)\mathrm{}$, and consequently the "renormalized" single-quasiparticle energy ${\epsilon }_p$, in terms of both scattering phase shifts and the Møller operator. The theory is compared with the Hartree approximation, the results of Allen, Nicolis et al., the work of Baerwinkel and Grossmann, and with expressions from Fermi-fluid theory.

An approximate variational calculation for the correlation energy of an electron gas is carried out using a Jastrow wave function and a linked-cluster expansion previously derived. The terms included in the variational calculation are the lowest-order direct terms in the potential and kinetic energies, and corresponding ring diagrams that incorporate the long-range screening effect, the lowest-order exchange kinetic energy, the lowest-order exchange potential energy and associated ring diagrams, and second-order ladder diagrams for the potential energy. The second-order energy terms are small compared with the first-order terms, suggesting that the expansion is valid. The pair correlation functions between electrons of parallel and antiparallel spin are also calculated and are non-negative for all $r$. The pair correlation function for electrons of parallel spins satisfies the condition $g\left(0\right)=0\mathrm{}$. The calculation was performed for different correlating factors between electrons of parallel and antiparallel spin, but this had a negligible effect on the correlation energy.

The magnetohydrodynamic drift-kinetic equation has been derived by means of the small-gyroradius expansion. The small-gyroradius expansion derived here employs the magnetohydrodynamic ordering. Using this small-gyroradius expansion, all the relevant transport coefficients are systematically evaluated. These results include both diagonal and cross coefficients for the particle fluxes, heat flux, and electric current. By combining the transport coefficients with appropriate moments of the drift equation, a closed set of equations which accurately summarizes the significance of these equations, in particular with regard to recent axisymmetric reactor experiments, provides an extension of the Chew-Goldberger-Low equations, now valid when the thermal velocity is comparable to the electric drift. The generalized equations may be used to investigate the macroscopic stability theory of doublet plasmas.

The Townsend primary ($\alpha$) and secondary ($\gamma$) coefficients and Paschen curves for neon, xenon, and their mixtures are reported. The primary coefficient was determined by measuring the variation in the luminous flux in a self-sustained Townsend discharge between two parallel-plate nickel electrodes spaced 21.5 mm apart. The values of the reduced primary coefficient, $\frac{\alpha }{p_0}$, for Ne and Xe are about 10-20% lower than published values obtained by the classical method of Townsend. The $\frac{\alpha }{p_0}$ values obtained for Ne agree with previous work using the same luminous-flux method. The ionization efficiency function $\eta$ $\eta =\frac{\alpha }{E}$, where $E$ is the electric field) has a maximum value for Ne and 0.01% Xe. Also, the value of $\eta$ for this gas mixture is larger than that for Ne + 0.1% Ar, especially at small values of $\frac{E}{p_0}$ (1-10 V ${\mathrm{cm}}^{-1\mathrm{}}$ ${\mathrm{Torr}}^{-1\mathrm{}}$).

We consider the problem of calculating the dynamic structure function $S(k,\omega )$ via a properly posed initial-value problem for a linear kinetic equation. A variational principle is introduced and shown to give a direct bounded estimate of $S(k,\omega )$. The variational trial functions for the problem are the space-time transforms of the phase-space correlation function. The choice of such trial functions are discussed and general expressions for $S(k,\omega )$ are developed which are applicable to simple gases. Numerical results are presented for the specific case of hard spheres, these results being compared to previous first-principle calculations based upon kinetic equations with and without the inclusion of memory effects. Excellent agreement with previous results, at all ratios of fluctuation wavelength to mean free path, is obtained with considerably less computational complexity. This computational efficiency of the variational approach suggests that it should be a valuable technique, from the standpoint of feasibility, in attempts at first-principle calculations for more complex many-body systems. In this regard, further study and improvement of the variational techniques themselves appear to be warranted.

Hydrodynamic boundary conditions are often described in terms of a slipping length which linearly relates the tangential velocity and its normal derivative at the boundary. In this paper a generalized wave-number and frequency-dependent slipping length is defined in terms of the eigenmodes of the nonlocal hydrodynamic equations. The ordinary slipping length is the zero-frequency, long-wavelength limit of this generalized slipping length. A scattering-theory formalism provides the connection between the eigenmodes and the generalized slipping length. The mode-mode coupling theory of a compressible fluid in the presence of a plane wall is developed. This theory indicates that the zero-frequency generalized slipping length diverges in the long-wavelength limit. The contribution due to sound-mode coupling dominates and diverges as $k^{\frac{-1\mathrm{}}{2}}$. While this result shows the nonexistence of the ordinary slipping length, it does not invalidate the use of no-slip boundary conditions in ordinary hydrodynamics. Corrections to Stokes's law are discussed in terms of the divergence of the slipping length, and the form of the divergence at low densities is indicated.

In dealing with critical phenomena of complex systems that simulate realistic materials, the full structure of the renormalization group is often unnecessarily cumbersome. For approximate calculations and for systems with special properties, specialized generators are simpler to apply. We derive several such exact and approximate differential generators and solve a number of interesting practical problems to illustrate this approach: (i) We derive a new approximate differential generator based on the Wilson incomplete-integration generator. Using this generator we calculate for an $n$-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order $\mathcal{O}$ and "propagator exponent" $\tilde{\sigma }$ to first order in the expansion parameter ${\epsilon }_{\mathcal{O}}\left(\tilde{\sigma }\right)=d+\mathcal{O}(\tilde{\sigma }-d)$; this extends previous work for $\tilde{\sigma }=2\mathrm{}$, $\mathcal{O}$ arbitrary; $\tilde{\sigma }\le 2$, $\mathcal{O}=2\mathrm{}$ (long-rang forces); and $\tilde{\sigma }=4\mathrm{}$, $\mathcal{O}=2\mathrm{}$ (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\tilde{\sigma }=2L$ ($L$ a positive integer ≥2) describe the onset of helical ordering for which $\left|\overrightarrow{\mathrm{k}}\right|\sim {\mathrm{}(-p)\mathrm{}}^{{\beta }_k}$, where ${\beta }_k=\frac{1}{2(L-1\mathrm{})\mathrm{}}+o\left({\epsilon }_{\mathcal{O}}^2\mathrm{}\right(2L\left)\right)$ and $p$ parametrizes the hypersurface of critical points. For $p>\mathrm{}0\mathrm{}$, the ordered phase is uniform; for $p<\mathrm{}0\mathrm{}$ there is spiral order. The point $p=0\mathrm{}$, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character $L$. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\to \infty$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\epsilon$ with $\epsilon \equiv {\epsilon }_2\mathrm{}\left(2\right)=4-d$.

The full eikonal approximation has been used to calculate differential and total cross sections for the elastic scattering of 50-, 100-, and 200-eV positrons from atomic hydrogen. The eikonal results are compared with other theoretical calculations and with the corresponding electron scattering cross sections. The electron and positron cross sections are found to differ as expected, in contrast to the Born and Glauber approximations which predict identical cross sections for both electron and positron scattering. Although no experimental results are available for comparison with our calculated cross sections, our results indicate that the full eikonal method may be useful for the calculation of positron (or proton) -atom scattering cross sections.

The lifetime of the ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ state of Sr has been measured by observing the intensity of the $5s^2{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}\leftrightarrow 5s5p{}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ fluorescence as a function of time, following the excitation of the ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ state by a 1-μsec pulse from a flashlamppumped dye laser. The measured value of the ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ lifetime is 21 ± 1 μsec.

The one-electron contributions to the hyperfine constants for the excited $S$ states of rubidium and cesium are calculated and compared with recent experimental results. From this comparison, the expected many-body contributions to the hyperfine constants are derived and shown to systematically decrease for higher excited states, in agreement with the situation found in earlier work on sodium. The physical origin of this behavior of the correlation effect is discussed.

The Salzberg-Prager-May equation of state is proved rigorously for two-dimensional rectilinear vortices moving in a finite solid container, in the microcanonical ensemble formulation.

It is argued that recent results of Showalter and Kay show a reasonable agreement (20%) with recent work of our group concerning the absolute values of the excitation cross sections. However, the analysis of the measured data performed by Showalter and Kay may introduce uncertainties into their triplet excitation functions, especially at higher energies.

We report the results of an analysis of the available experimental data on electron-induced $K$-shell ionization which show that the data can be described by the Bethe theory of ionization corrected for relativistic energies by a classical correction. This analysis, which extends over five orders of magnitude in the ratio of bombarding energy to ionization potential, appears to be superior to the available theoretical treatments, and suggests where further work would be useful.