Uncertainty principles for the average fluctuations of momentum and position per particle in an ensemble are derived. For bosons, these results are similar to the usual uncertainty principle, with slight corrections for the correlated motion of the particles. For fermions, a large correction for the Pauli exclusion principle is necessary, which causes $〈p^2〉〈{\mathcal{r}}^2〉$ to be proportional to the two-thirds power of the number of particles in the ensemble.

A relativistic expression for $L_1{L}_2{M}_{4,5}$ Coster-Kronig transition probabilities has been obtained by using the solutions of the Biedenharn symmetric Dirac-Coulomb Hamiltonian. The interaction matrix element yields the well-known nonrelativistic result in the appropriate limit. Numerical results for atoms having $Z$ values between 32 and 41 are compared with earlier calculations. Relativistic effects are shown to be sufficiently pronounced as not to be critically affected by the choice of a screening parameter. The transition probability at the cutoff point is appreciably greater than earlier nonrelativistic values.

The optimum scale factor for the Slater free-electron exchange approximation for atoms has been obtained by minimizing the total electronic energy. In general, the scale factor lies between the Slater and the Gaspar-Kohn-Sham limit of 1 and $\frac{2}{3}$, respectively. The variation that occurs as the different shells are populated is qualitatively understandable in terms of the inhomogeneity of the over-all charge distribution.

Electron transport is considered in the metallic range of metal-ammonia solutions; the scattering centers are (i) ions and ammonia dipoles weakly bound in solvated ion complexes, and (ii) free ammonia dipoles. They form a binary liquid mixture, but only one of the three possible partial structure factors is required. From the orientational degrees of freedom of the dipoles, it can be shown that much of the scattering is incoherent. This, combined with the observation that solvated ions are relatively weak scatterers, accounts in large measure for the high conductivities measured in metal-ammonia solutions.

The influence of charged-particle interactions on the photoionization cross section of ${\mathrm{H}}^{-}$ is reinvestigated. In particular, polarization of the initial state which gives rise to a free $s$-wave electron in the final state is considered. It is found that a perturber density of roughly ${10}^{20}$ ${\mathrm{cm}}^{-3\mathrm{}}$ is necessary in order for this process to compete with the usual transition in the near-infrared region. This is in sharp disagreement with the result of a previous investigation.

The polarized-orbital method and many of the common variations and approximations of this approach are tested in a computation of the binding energy of the negative hydrogen ion ${\mathrm{H}}^{-}$. It is found that if a trial wave function for ${\mathrm{H}}^{-}$ is constructed from a distorted atomic wave function in which the first-order perturbed orbital is simply added to the undistorted atomic function, the result is a very poor approximation (binding energy -0.0094 Ry). However, if the trial function is modified, as suggested by Drachman, with the introduction of an additional independent function multiplying the perturbed orbital, the result is quite good, the binding energy being -0.0544 Ry. For comparison, the exact value (due to Pekeris) is -0.0555 Ry.

The localization of electronic states in one-dimensional disordered systems is examined in terms of the reflection and transmission coefficients. The transfer-matrix method is used. The main body of the work deals with a one-dimensional liquid model in which the central part of the potential remains the same in all cells, and only the lengths of the flat arms vary from cell to cell. It is found that the contribution of the initial phase of a wave at the zeroth cell to the phase at the $n\mathrm{th}$ cell is reduced by a factor $\frac{\mathrm{}(1-|\mathrm{}\mathcal{r}\mathrm{}\left|\right)\mathrm{}}{\mathrm{}(1+|\mathrm{}\mathcal{r}\mathrm{}\left|\right)\mathrm{}}$ every time in passing through a cell. When the phase memory is completely lost, ${\Phi }_j\sim {\phi }_j$, where the reflection coefficient of the $j\mathrm{th}$ cell is ${\mathcal{r}}_j=\left|\mathcal{r}\right|e^{i{\phi }_j}$. If ${\Phi }_j$ obeys a uniform or nearly uniform probability distribution, the wave function always grows exponentially. It is shown that in most cases, especially when cell size distribution has a wide spread, $P\left(\Phi \right)$ is nearly always uniform. All wave functions are localized in a completely disordered system, but in the one-dimensional liquid model nonlocalized states do exist.

This paper investigates the classical motion of oscillator systems governed by Hamiltonians having the nearly linear form $H=\Sigma \stackrel{N}{k=1\mathrm{}}\frac{{\omega }_k}{2}({P_k}^2+{Q_k}^2)+\gamma (V_3+V_4+\cdots ),$ where $N$ is the number of oscillators, ${\omega }_k$ are the positive frequencies of the harmonic approximation, $\gamma$ is the nonlinear coupling parameter, and $V_3$, $V_4$, etc., are cubic, quartic, etc., polynomials in $Q_k$ and $P_k$. The purpose of this investigation is to demonstrate that macroscopic irreversibility is an inherent property of physical, nearly linear oscillator systems even in the limit as $\gamma$ tends to zero. Irreversibility occurs for these systems because of the appearance of resonance overlap, which causes the system trajectories to wander more or less randomly over part or most of the energy surface. Resonance overlap can occur for arbitrarily small but nonzero $\gamma$ provided that $N\ge 3$ and that the ${\omega }_k$ satisfy commensurability conditions which allow the $V_3$ and/or $V_4$ interaction terms to resonantly couple all internal degrees of freedom. These results are demonstrated through an extensive computer study of the case $N=3\mathrm{}$. This case is especially suitable for study since it possesses much of the complexity of the full many-body problem and yet is sufficiently simple to yield to level-curve analysis which provides an especially lucid pictorial display of the random motion of individual trajectories. In addition, the computer study shows that one consequence of resonance overlap is that system trajectories originally close to each other in phase space can move apart more or less exponentially with time. This exponential stirring of phase space is of the type which Gibbs envisioned as leading to irreversible behavior. In particular, the slightest uncertainty of the initial state leads very quickly to complete uncertainty of the final state. Moreover, resonant oscillator systems share ownership of exponentially divergent trajectories with gaseous systems. Indeed, this property, which is a direct consequence of the equations of motion, is perhaps the ultimate source of irreversibility.

A translational band model is applied to the calculation of adsorption isotherms of noninteracting fermions and bosons adsorbed on a crystalline surface. The theory yields analytic expressions for the vapor pressure as a function of temperature and monolayer coverage, in terms of the band parameters of the adsorption system. In the case of fermions, the actual band widths and band gaps can be mapped by a low-temperature isotherm, which will display multiband steps analogous to the multilayer steps of interacting adatoms. The significance of "mobility" and "localization" is discussed, and it is argued that the words have physical meanings only in terms of comparisons between the dwell time of an atom on a specific site and a characteristic time of the experiment in question. In the case of adsorption isotherms, the experimental time is related to the vapor pressure, and the band theory offers a criterion for mobility in terms of the energy spectrum of the adsorbed atoms. Numerical estimates indicate that helium monolayers satisfy the criterion for mobility at 4 °K and below, but that Ar and ${\mathrm{N}}_2$ monolayers may be localized at 77 °K. The isotherms of all physically adsorbed systems satisfy the criterion for mobility at sufficiently low temperatures. In contrast to the heat capacity, isotherms in the localized and mobile regimes are not qualitatively different. The effects of interactions among the adatoms and of surface inhomogeneities are briefly discussed.

A theory is presented to predict the size of the strain field about an isotopic impurity in solid helium and neon. The theory is based on the self-consistent harmonic approximation, as recently formulated by Koehler. In contrast to previous treatments, the theory computes the long-range behavior of the strain field and scattering rates for long-wavelength phonons. Good qualitative agreement with experimental results is found for the pressure dependence of the scattering rate for helium, but the quantitative agreement is dependent on the assumptions used in determining parameters of the theory. The theory predicts the existence of an appreciable strain field in neon.

A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^4$, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.

Using close-coupling calculations of Burke and Moores for the scattering of electrons by ${\mathrm{Mg}}^{+}$ ions in the $3{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}$ and $3{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2},\frac{3}{2}}^{}{}_{}{}^{}$ states, Baranger's expression for the impact approximation width of an isolated line is implemented for the components of the resonance doublet. These widths are extrapolated to below inelastic thresholds and averaged over elastic resonances according to theoretical threshold laws. In the experimental energy range, results compare reasonably with semiclassical approximations and with a semiempirical method involving effective Gaunt factors extrapolated to zero electron energy.