Differential and total cross sections for $e$-${\mathrm{H}}_2$ collisions have been calculated by the close-coupling procedure using three different model interaction potentials. Particular attention is focused on the simplest possible model having the desired physical properties. This model uses square wells to describe short-range interactions and the known polarization and quadrupole interactions for long-range potentials. The results obtained for this square-well model for $e$-${\mathrm{H}}_2$ collisions are very encouraging considering its simplicity. Comparisons with results obtained using other model potentials are presented.

The theory of effective potentials is developed in a way which lends itself to variational formulations. The positron-helium and electron-helium systems are chosen to illustrate the method. The effective potential is defined in terms of the resolvent operator for a system which has been modified by removal of the open-channel states. Rigorous maximum and minimum principles are derived which should be useful in the computation of the effective potential and which are valid, for sufficiently low energies, even when the target bound-state wave function is not known exactly. A numerical application of this method to positron-hydrogen scattering has been made and results are reported. The effective-potential approach to the scattering problem leads naturally to a model for resonance calculations. This model is reviewed here and formulated variationally. An Appendix is devoted to further elaboration of this approach to resonance calculations in the context of potential scattering, with the aid of Jost-function theory.

A model Hamiltonian is used to find the dispersion relation of trapped rotons under the influence of a vortex field by a variational technique. The lowest branch corresponds to the energy gap ∼ 2 °K, in good agreement with the result obtained by previous semiclassical theory. The trapped mode explains the recent mobility measurements of trapped ions well.

This paper deals with the equations of motion of the density matrix of a two-level atom in the presence of an intense multimode radiation field characterizing a multimode laser. Using a simplified treatment of line broadening which does not include correlations between momentum-changing collisions and pressure-broadening collisions, we obtain expressions which show the effect of pressure, laser-pulse length, and intensity on the excitation. We obtain analytic expressions in limiting cases of the pressure which allow prediction of the degree of excitation from a given laser pulse. In our treatment the phase relaxation time of the atoms is assumed to be fast enough so that coherence effects among the atoms can be ignored. Treating the velocity of the atoms statistically, we solve for three cases: constant velocity, velocity changes that are fast compared to the excitation time but slow compared to the dephasing time, and velocity changes that are fast compared to all other processes. The three cases yield the same formal equation for the two-body process of absorption of photons by incoherent atoms; however, the cross sections differ and are calculated explicitly for each case. Also discussed are the range of laboratory conditions and the relationship between these laboratory conditions and the various cases considered to demonstrate that under most circumstances this simple photon-absorption picture is applicable, provided the cross section is calculated correctly.

We report systematic measurements of the nuclear-spin-lattice relaxation times $T_1$ in the ordered phase of solid hydrogen. The dependence of $T_1$ on temperature $T$ and on the ortho content $X$ is found to be described by $T_1=A{X}^4{T}^{-1\mathrm{}}{e}^{\frac{B}{T}}$, where $A=1.0\mathrm{}\pm 0.3$ min, $B\simeq \mathrm{}[0.4(1.8X-1)+0.24]°$K for $X>\mathrm{}0.56\mathrm{}$ and $B\simeq 0.24°$K for $X<\mathrm{}0.56\mathrm{}$. This functional dependence is in qualitative agreement with that calculated from the rotational correlation times of the quantum crystal lattice. The applied field was 75 kG and the temperature range investigated was 0.075 to 1.5 °K.

The inelastic-scattering functions for liquid argon have been measured at 85.2 K. The coherent-scattering function was obtained from a measurement on pure ${\mathrm{A}}^{36}$ and the incoherent-scattering function was derived from the result obtained from the ${\mathrm{A}}^{36}$ sample and the result obtained from a mixture of ${\mathrm{A}}^{36}$ and ${\mathrm{A}}^{40}$ for which the scattering is predominantly incoherent. The data, which are presented as smooth scattering functions at constant values of the wave-vector transfer in the range 1.0-4.4 ${\mathrm{\AA }}^{-1\mathrm{}}$, are corrected for multiple-scattering contributions and for resolution effects. Such corrections are shown to be essential in the derivation of reliable scattering functions from neutron-scattering data. The incoherent-scattering function is compared to recent molecular-dynamics results and the mean-square displacement as a function of time is derived. The coherent-scattering function is compared to molecular-dynamics results and also, briefly, to some recent theoretical models.

The Stark-broadened profiles of the Balmer lines ${\mathrm{H}}_{\alpha }$, ${\mathrm{H}}_{\beta }$, ${\mathrm{H}}_{\gamma }$, and ${\mathrm{H}}_{\delta }$, have been measured in a high-current, wall-stabilized arc operated in hydrogen. Temperatures and electron densities have been determined with a plasma model assuming partial local thermodynamic equilibrium (LTE), after a detailed plasma analysis revealed small deviations from complete LTE. The temperatures have been determined from line-to-continuum intensity ratios using ${\mathrm{H}}_{\beta }$ and continuum points in the near uv, and the electron densities were derived from absolute intensity measurements. The investigations cover the range of electron densities between 1.5 × ${10}^{16}$ and ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and temperatures between 0.9 × ${10}^4$ and 1.4 × ${10}^4$ K. Extensive comparisons with recent Stark-broadening theories and other experiments have been undertaken with the following principal results: The most pronounced differences between this experiment and theory occur in and near the line centers, where the measured profiles show systematically less structure than the calculations predict. Comparisons of the calculated and precisely measured ratios between the ½, ¼, and $\frac{1}{8}$ widths within each line show that the recent theories are internally consistent within 6% for ${\mathrm{H}}_{\beta }$ and ${\mathrm{H}}_{\gamma }$, whereas for ${\mathrm{H}}_{\alpha }$ inconsistencies of order 25% occur. Similar inconsistencies, somewhat larger respectively, are found for the theoretical values of the half-width ratios between different Balmer lines. For the most important line ${\mathrm{H}}_{\beta }$, the measured half-widths agree within 7% with the calculated values. The total experimental error in this comparison, which originates predominantly in the electron-density measurement, is estimated not to exceed 6%. Very reproducible asymmetries and red shifts are observed for ${\mathrm{H}}_{\beta }$ and ${\mathrm{H}}_{\gamma }$. Somewhat smaller red shifts are also obtained for ${\mathrm{H}}_{\alpha }$. The shifts are approximately linear functions of the electron density. Comparisons with other experimental data show appreciable scatter between the various results. We estimate that for the most-often-studied line ${\mathrm{H}}_{\beta }$ the uncertainties in the theoretical Stark widths are of the order of (5-7)% for the range of our experiment.

Molecular dynamics of hard parallel s$\stackrel{\mathrm{´}}{\mathrm{q}}$uares subject to periodic boundary conditions appear to lead to a first-order phase transition with a van der Waals loop (at least for a system of 400 particles). The density change across the transition is small (about 2%). Results on the self-diffusion and velocity autocorrelation function $\rho \mathrm{}\left(t\right)\mathrm{}$ for 400 and 900 particles are also presented. $\rho \mathrm{}\left(t\right)\mathrm{}$ decays exponentially for short times, and at least in the intermediate fluid density, decays at $t^{-1\mathrm{}}$ for long times $t$. In the solidlike region anticorrelating effects predominate.

Electrostrictive laser-beam self-focusing was simulated on a digital computer according to my previous derivation. The results show the beam focusing itself into a long thin filament a few wavelengths of light in radius, propagating with nearly constant radius up to several centimeters, and diverging again. The filament radius decreases throughout the pulse. It also decreases with increasing pulse power. The filament length varies approximately as the square root of the input power above threshold. Growth upstream and downstream is initially at ${10}^7$ to ${10}^8$ cm/sec. The maximum extent is reached before the peak of the incident laser pulse, but the maximum intensity is achieved after the peak. Computer "streak diagrams" are very similar to recent streak photographs of track formation. The filamentary theory is more attractive than the moving-focus theory presented previously as no rapidly responding (fast) self-focusing mechanism is required to explain track-growth spurts with rapidly temporally modulated pulses. The trapping threshold for an arbitrary combination of fast and electrostrictive self-focusing mechanisms is presented. The predominance of a fast mechanism would contradict the experimental variation of the trapping threshold with initial beam-waist radius. The results suggest much of the nonlinear response previously measured by others for BK-7 glass is caused by electrostriction rather than fast mechanisms.

A number of radiating two-level systems subject to an external resonant driving field are treated as a single angular momentum system. It is shown that the damping of the oscillation of the energy expectation value is the result of quantum-mechanical averaging over an ensemble of experiments, the energy oscillation in a single experiment being undamped.

The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy $\hslash \omega L_3$, that describes the collective effect of a number of identical two-level (or spin-½) systems under given idealized conditions, a large total angular momentum quantum number $L_0$ corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay—with the NO initially excited and the radiation field in the ground state—and forced oscillation—with the NO subject to a prescribed resonant driving field—are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large $L_0$, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as $L_3\mathrm{}\left(0\right)\mathrm{}$ approaches $L_0$ (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to $L_3\mathrm{}\left(t\right)\mathrm{}$. It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of $L_3\mathrm{}\left(t\right)\mathrm{}$ to a constant (negative) value—or steady state—at which the power absorbed equals the power radiated; in the strong-field case, $L_3\mathrm{}\left(t\right)\mathrm{}$ oscillates periodically between the limits $\pm L_0$, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for $L_0=\frac{1}{2}$, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=-L_0, 〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For $L_0>\frac{1}{2}$, the free decay is treated by a method that involves the derivation of a set of expressions for the $k\mathrm{th}$ derivative of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ as an expectation value of a polynomial in $L_3$ of order $k+1\mathrm{}$. This set, together with the eigenvalue equation for $L_3$, is shown to lead to a solution for all the moments of $L_3$. The method is used to obtain complete solutions for several low values of $L_0$, and also to obtain initial derivatives of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ as polynomials in $L_0$ for $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=L_0$. For large $L_0$, a comparison of classical and quantum-mechanical equations shows that only the condition $〈L_3\mathrm{}\left(0\right)\mathrm{}〉\approx L_0$ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition $〈L_3\mathrm{}\left(0\right)\mathrm{}〉=L_0$ solved quantum mechanically up to such a time $t_1$ by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ge t_1$. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$—for arbitrary $L_0$—is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of $L_0$. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and $〈L_2^3\mathrm{}\left(t\right)\mathrm{}〉$ is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since $〈L_3\mathrm{}\left(t\right)\mathrm{}〉$ describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.

A quantum theory of diffusion is presented and applied to the diffusion of isotopic impurities in solid helium. For temperatures much less than the Debye temperature $\Theta$ and much more than the impurity exchange temperature $\frac{\hslash J}{k_B}$, it is shown that the diffusivity is given by $D=\left(\frac{J{a}^4}{{\sigma }^{*}x}\right)$. The effective cross section ${\sigma }^{*}$ for the scattering of two mobile impurity atoms is of the order of a square lattice spacing $a^2$, and the mole fraction $x$ of the impurity atoms is assumed to obey $\left(\frac{\hslash J}{k_B\Theta }\right){\left(\frac{T}{\Theta }\right)}^7\ll x\ll 1$. Observation of the concentration dependence $D\propto \frac{1}{x}$ would constitute strong evidence of quantum mobility, which has been of considerable theoretical interest in recent years.

A theoretical calculation of the stimulated thermal scattering of picosecond laser pulses by common liquids is presented. The calculation is made for the experimental configuration of Mack. The behavior of the first-order diffraction spots is studied in detail by varying liquid and laser parameters. A comparison with Mack's observations is made.

The transient phase-independent solutions of the laser Fokker-Planck equation are analyzed for a short range of time following the moment of the switching on of the oscillations. The evolution of the laser light in time is then represented as a nonstationary Gaussian random process. On the basis of this fact, the factorial moment generating function of the photocount probability distribution is derived, which accounts automatically for all the finite-counting-time corrections. The time development of the first two moments is compared with the recent results of Arecchi and Degiorgio.

To take into account the collective aspects of motion of liquid atoms, an expression for the memory function of a classical liquid has been formulated from the equations for the velocity autocorrelation function by using the approaches of Kerr and Singwi et al. for atomic motion in classical liquids. This expression satisfies the moment relation, and its long-time behavior shows a $t^{-\frac{3}{2}}$ dependence. Furthermore, approximate calculations for the memory function in liquid argon are made by using our expression and incorporating the contribution due to the hard-core part of the potential from the expression of Singwi et al. These calculations show the presence of a minimum in the memory function near 4 × ${10}^{-13\mathrm{}}$ sec.

The ground state of an electron gas is studied by means of the method of correlated basis functions developed by Wu and Feenberg. The optimum form of the Bijl-Dingle-Jastrow-type wave function of a charged-boson gas obtained with the application of the paired-phonon analysis is used as the correlation factor in a trial wave function describing the electron gas. Numerical results are given for the correlation energy per particle and the radial distribution function at metallic densities. The high-density limit of the system is briefly discussed.

Some consequences of the Percus-Yevick theory are studies in the neighborhood of the critical point for adhesive hard spheres and for the 6:12 potential (truncated at $6\sigma$). It is shown that the Percus-Yevick theory gives rise to classical behavior at the critical point. In particular, it is shown that for the compressibility equation of state the critical exponents $\gamma$ and $\delta$ are 1 and 3, respectively, and for the energy equation of state the critical exponents $\alpha$ and $\beta$ are 0 and ½, respectively. In addition, the behavior of the Percus-Yevick distribution function in the neighborhood of the critical point is examined and it is shown that for the critical isochore the temperature derivative of the distribution function diverges with a critical exponent of ½ which is independent of $\mathcal{r}$ and that for the critical isotherm the distribution function is a linear function of the density for all $\mathcal{r}$.

The retarded van der Waals potential between pairs of spinless atoms, H-H, H-He, He-He, Ne-Ne, Ne-Ar, and Ar-Ar, was computed with covariant formulas of Feinberg and Sucher. Electric and magnetic effects were treated on equal footing. For hydrogen, which posses-a large magnetic polarizability, magnetic effects dominate for large distance of separation, and for small distances, an additional contribution to the London constant for a pair of hydrogen atoms from the magnetic effects is found to be $0.0155{{\alpha }_a}^5$, about 0.24% of the currently accepted value.

It is shown that the static dielectric function computed from a variational solution of the appropriate vertex function obeys the compressibility sum rule and reaches the well-established limit for large $\overrightarrow{\mathrm{k}}$. The vertex equation contains the self-energy contributions to the one-electron states as well as the exchange contributions. A comparison of the results with a recent scheme due to Toigo and Woodruff shows that the moment-conserving approximation used by these authors is correct for small wave vectors and incorrect for large wave vectors in contrast to the variational method.

In this paper we examine the treatment of the Fermi shift in the paper by M. H. Choudhury, Phys. Rev. 186, 66 (1969). Some difficulties are pointed out which render Choudhury's results to be of doubtful value. It is seen that starting from Choudhury's Hamiltonian one may recover an equation considered earlier by Mittleman for studying the Fermi-shift phenomenon. Finally, the Mittleman result is employed to make new estimates for the electron-neutral-atom scattering lengths.