This paper investigates whether it is possible to justify from first principles of quantum mechanics, the methods which, as in Dewar's split-$p$-orbital and Löwdin's alternate-molecular-orbitals method, suppose that in some regions of space the wave function is zero. The answer is affirmative, and a demonstration is given using some properties of the coordinate operator. The energy of the system may be expressed as a sum of certain integrals which are not carried out on the whole space but only on some closed intervals. These intervals may be found via the variational principle.

Cross sections are reported for production of positive ions and electrons in collisions of hydrogen atoms in the energy range 1-25 keV with He, Ne, A, Kr, Xe, ${\mathrm{H}}_2$, and ${\mathrm{O}}_2$. Slow positive ions are produced by capture of an electron by the fast atom from the target gas and by ionization of the target. Electrons are produced by target-gas ionization and by stripping of the fast atom. Ionization cross sections for each of the target gases are derived from the positive-ion production cross sections and previous measurements of the capture cross sections.

A general method is described of building up open-shell multideterminant wave functions which are eigenfunctions of the spin operators $S^2$ and $S_z$. In the present case, that of a molecule with an Abelian symmetry group, the wave functions obtained are easily adapted for spatial symmetry. The matrix elements of the Hamiltonian are derived. The method gives a building up of the matrix elements of the irreducible representation [$n_1,n_2$] symmetric groups which, as far as the author knows, is original. While many of the specific results are not new, the author believes that the presentation which he used is both new and interesting. This method will be extended later to more general cases.

Measurements on a 60-kV 1-μF vacuum spark discharge indicate the following sequence of events. During the early stages of the discharge, the current is carried mostly by electrons which are accelerated to the full tube voltage, as seen from intense bremsstrahlung x-ray production at the anode. The fast electron current and accompanying x-ray emission then diminish, but the total current continues to increase. During the time of reduced x-ray emission, the plasma emits a strong microwave pulse which peaks near the frequencies characterizing the electron drift instabilities. The dense beam of fast electrons then reappears for about 50 nsec and the whole cycle repeats, perhaps twice more. Although the fast electron stream repeatedly appears and disappears, the total current (fast stream plus drift current) usually rises monotonically. When a current of about 4×${10}^4$ A is reached (in ∼5×${10}^{-7\mathrm{}}$ sec), dense material vaporized from the anode reaches the cathode and the tube impedance collapses. These events are explained in terms of a model which treats the discharge as alternating between two current modes, one dominated by a fast electron beam, the other by electron drift instabilities.

The equilibrium thermodynamics of a one-dimensional system of bosons with repulsive $\delta$- function interaction is found to be intermediate between those of a one-dimensional free-boson and a one-dimensional free-fermion system. Numerical comparisons are given.

The thermal-expansion coefficient of liquid helium II under its saturated vapor pressure has been measured from 0.85 K to within 0.4 mK of the $\lambda$ transition. The capacitive technique employed was well suited for these measurements. With the aid of the Clausius-Mossotti equation, we could resolve a relative change in density of 1 part in ${10}^6$ in a manner free from sources of systematic error. The data below 1.60 K were analyzed to obtain information about the density derivatives of the Landau parameters $\Delta$, $p_0$, and $\mu$ with the following results: $\frac{\rho }{\Delta }\frac{\partial \Delta }{\partial \rho }=-0.\mathrm{}763\pm 0.012 \mathrm{and} \frac{1}{2}\frac{\rho }{\mu }\frac{\partial \mu }{\partial \rho }+2\mathrm{}\frac{\rho }{p_0}\frac{\partial p_0}{\partial \rho }+\frac{\rho }{\Delta }\frac{\partial \Delta }{\partial \rho }=-1.638\mathrm{}\pm \mathrm{0.077.}$ The results of a reanalysis of the data obtained by other workers in this temperature interval are presented in order to provide a valid comparison with the present results. The techniques employed also allowed measurements to be taken to within 0.4 mK of the $\lambda$ transition. At this temperature, the uncertainty in an individual measurement (one standard deviation) had reached about ± 10%. This represents a considerable improvement over previous measurements and allows a detailed examination of the validity of the Pippard-Buckingham-Fairbank relations. It was found that the agreement between theory and experiment is good if the temperature dependence of all the terms in these relations is taken into account. The capacitance bridge was also employed to measure absolute values of the dielectric constant of liquid helium at temperatures of approximately 1.0 K and a frequency of 5 kHz. Analysis of these measurements yielded a value for the molar polarizability of 0.123 296 ± 0.000 030 ${\mathrm{cm}}^3$ ${\mathrm{mole}}^{-1\mathrm{}}$.

We have completed a remeasurement of the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-2\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ energy separation in atomic hydrogen, using an atomic beam microwave technique. The final result obtained from measurements on four resonances is 9911.173 ± 0.042 MHz. When this is combined with the recent remeasurement by Robiscoe of the $2{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-2\mathrm{}{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ energy separation, we obtain a value for the fine-structure splitting $\Delta E$ and the fine-structure constant ${\alpha }^{-1\mathrm{}}=137.035\mathrm{}8\left(5\right)\mathrm{}$.

In this paper, we consider the photoelectron counting distribution when the wave field incident on the detector is a mixture of harmonic signal and thermal noise. An expression for the factorial moment-generating function of the number of counts is obtained in terms of the eigenvalues of an associated integral equation. This expression is then used to evaluate the factorial cumulants of the number of counts. The results are valid for an arbitrary counting interval. The extension to photoelectron counting with several detectors is considered in paper II of the present investigation.

The rate constant for the transfer of a ${\nu }_3$ quantum from C${\mathrm{O}}_2$ to ${\mathrm{N}}_2$O is calculated using a second-order formulation of the theory of energy transfer due to long-range forces. This formulation is valid if ${\omega }_{\mathrm{vi}}\tau >1\mathrm{}$, where ${\omega }_{\mathrm{vi}}$ is the frequency (rad ${\sec }^{-1\mathrm{}}$) for transition from initial to virtual state, and $\tau$ is the time duration of the collision. The theory is estimated to be accurate to within a factor of 2 and agrees with the room-temperature experimental results. The formulation is also generalized to include energy transfer due to Debye induction interactions.

Approximate equations of motion for the single-particle autocorrelation function $f(\overrightarrow{\mathrm{k}},\overrightarrow{\mathrm{p}},{\overrightarrow{\mathrm{p}}}^{\prime },t)$ in a classical fluid are considered. The interaction term is treated by expanding it in powers of the approximate collective coordinates of the system. In the detailed calculations, only terms linear in the collective coordinates are retained. Two models are considered using two sets of variational collective modes. If the isothermal sound mode introduced by Zwanzig is used, a more physically transparent derivation of the modified linear Vlasov equation, obtained earlier by several authors, results. The phononlike modes introduced by Nossal and Zwanzig give a new and significantly improved result which leads directly to closed expressions for the longitudinal and transverse current-current correlation functions. The models presented show a clear separation of structural, dynamic, and thermal motion effects when analyzed in terms of appropriately defined memory functions. Extensions of the model results which replace free-particle motion by the "self-motion" of an atom in the fluid are naturally suggested, though no satisfactory theoretical justification for these extensions is given. The extension of the modified linear Vlasov equation is the same as proposed earlier by Kerr. The extension of our phononlike model is a new result which is expected to give improved agreement with experiment.

A theory describing two sequential electromagnetic pulses interacting with a resonant, inhomogeneously broadened attenuator is presented. The resulting equations are solved on a computer. Limiting cases are considered which indicate the conditions under which the second pulse is expected to increase in energy. The self-induced transparency effect in a two-pass attenuator is treated using this theory. It is shown that there are qualitative as well as quantitative differences between the single-pass and double-pass cases. For example, curves of output versus input energy in the two-pass case may have observable structure that is sensitive to the presence or absence of atomic coherence. It is found that under a wide range of circumstances, the output for the second pulse is independent of whether the pulse is going in the same or in the opposite direction from the first. This result is seen to be related to more general considerations involving the interaction of a pulse with spatially inhomogeneous systems such as a chain of amplifiers and attenuators. Also discussed are the measurement of $T_1$ using two-pass techniques, and the possibility of using these phenomena in a practical application to optical logic circuitry.

The continued-fraction expansions in terms of Brillouin-Wigner perturbation energies are evaluated for the He-like series. They show remarkable convergence and stability properties. Both Padé approximants and perturbation energies are computed with formulas derived by the inner-projection technique.

A model is given that describes the quantum scattering of a ${\mathrm{He}}^3$ impurity from a rectilinear vortex in liquid He II; the principal assumption is the existence of an impurity wave function that satisfies a Schrödinger equation. Various possibilities for the impurity-vortex interaction are discussed. For certain interactions, the theoretical results are consistent with the experimental work of Rayfield and Reif provided spatial variations in the superfluid density are considered. A $T$-matrix formalism is also developed and applied to the vortex scattering of impurities, as well as phonons and rotons. The results are compared to existing theoretical and experimental work; the main discrepancies occur for roton scattering.

The high-temperature part of the fluid-solid coexistence curve for Lennard-Jones systems is investigated by "exact" Monte Carlo calculations. The interaction potential is separated into its repulsive ($\frac{4}{r^{12}}$) and attractive ($-\frac{4}{r^6}$) parts. The repulsive part is treated "exactly" by Monte Carlo computations with a 864-atom system, and the attractive part is treated as a perturbation. The "unperturbed" potential, which is homogeneous in the coordinates of the interacting particles, has trivial scaling properties which greatly simplify the computations. The attractive perturbation is treated to first order; the second-order corrections are shown to be very small at not-too-low temperatures. A high-temperature equation of state is obtained for the Lennard-Jones fluid, which is in excellent agreement with exact Monte Carlo computations at temperatures as low as about twice the critical temperature. Using the Hoover-Ree scheme, the free energy of the solid is determined and the transition densities and pressures calculated in the first-order approximation, which is shown to be quite satisfactory. The validity of Lindemann's melting "law" and a crystallization criterion based on the maximum of the structure factor are investigated.

An approximation scheme for a Bose liquid is presented, based on small fluctuations in density and currents. The method of obtaining the elementary excitations and their interactions in any order of the approximation is outlined. Also it is shown that the first-order calculations are in agreement with the calculations of Feynman, Bogoliubov, and Pitayevski. Second-order calculations agree with the improved results of Feynman and Cohen.

The semiclassical theory of gas lasers has been reformulated by adding rate terms to the density-matrix component differential equations. The solution to these equations, in the form of a Fourier series, is applicable at high laser intensities. A calculation of the effect of phase-changing collisions is also included so that the results can be compared to experimental data taken with a He-Ne laser operating at a wavelength of 1.15 μm.

A more powerful version of a numerical method published before is given which permits the integration of the Schrödinger equation for a large ensemble of particles. A new method for obtaining quantum expectation values is outlined. Numerical results are tabulated for the energy of the ground state of a system of 32 bosons interacting by Lennard-Jones potentials and with periodic boundary conditions. This is a commonly used model of liquid ${\mathrm{He}}^4$. The present results are substantially lower and closer to experiment than variational estimates based on a Jastrow-type wave function.

It is shown, by almost rigorous arguments, that there exist many-body states of a system of interacting bosons which exhibit both crystalline order and Bose-Einstein condensation into the zero-momentum eigenstate of the single-particle density matrix. The implications of this result are discussed in relation to theories of superfluidity and the nature of quantum crystals.

We examine the implication in the fact that, in a semiclassical theory of atomic structure proposed by Jaynes and Crisp, the initial state of an atom plays a critical role in determining its average lifetime in an excited state.