An effective interaction between atoms adsorbed on a substrate is calculated. The interaction arises from the exchange of phonons in the substrate and makes a density-dependent contribution to the ground-state energy per particle of the adsorbed system. Our results are applied to the submonolayer helium film adsorbed on argon-plated copper studied by Stewart and Dash. It is shown that, for this system the interaction is attractive and contributes to the ground-state energy per particle an amount of the order of - 1 °K, which is a significant fraction of the observed binding energy. The role of the effective interaction in bringing about the enhancement of the binding energy and the formation of dense clusters observed by Stewart and Sash is discussed.

The only if aspect of a quantum form of a Gibbs inequality is proved.

A pertubation method for determining the first quantum correction to the square-well classical second virial coefficient in a high-temperature approximation is presented. It is an alternative method to that presented by Nilsen, and agrees with his result to first-order terms in $\beta \epsilon$, where $\beta ={\mathrm{}\left(\mathrm{kT}\right)\mathrm{}}^{-1\mathrm{}}$ and $\epsilon$ is the depth of the well.

Recent indications that $R_4$ is a fairly good symmetry for two-electron atoms are shown to be unjustified.

The two different uses of $R_4$, one as a mathematical and one as physical group which were discussed in previous work, are presented. The treatment of Rau is shown to be equivalent to the mathematical approach. The use of the physical group leads to new relationships not present in the mathematical approach.

Additional correction terms of order $\frac{{\left(Z\alpha \right)}^{2}m}{M}$ are found for the electron $g$ factor in the ground state of hydrogenic atoms. This modifies earlier results given by Hegstrom and by this author.

We describe a new approach to the problem of superradiant spontaneous emission, which is applicable whether the number of atoms is large or small. We use this approach to formulate intensity correlations of arbitrary order, and then explicitly evaluate the second-order normally ordered relative intensity fluctuation. Large fluctuations at the onset of emission indicate the discrete nature of the spontaneous emission process; and the very small fluctuations found later at the emission peak show that the emitting process has by then become essentially classical.

The critical isotherm of ${\mathrm{He}}^3$ has been measured over a density range $\left|\Delta \rho \right|=\frac{|\rho -{\rho }_c|}{{\rho }_c}$ up to 0.6, and the chemical potential change $\mu -{\mu }_c$ was obtained as a function of the density change $\rho -{\rho }_c$. The departure from perfect antisymmetry about ${\rho }_c$, which can be represented by an expression of the form ${\left|\Delta \rho \right|}^{\varepsilon }$ with $\epsilon =6.2\mathrm{}$, is compared with theoretical suggestions.

The analysis of the preceding two papers in this series is applied to the specific case of a degenerate Bose fluid, in which there can be macroscopic occupation of the zero-momentum state (in a system at rest). The quasiparticle ensemble of Mohling and Tuttle is incorporated into the theory in order to guarantee that Nernst's theorem for the entropy is satisfied. It is shown that a double-quasiparticle model emerges naturally from the theory, as developed earlier, and this feature is then incorporated into the quasiparticle ensemble. The general theory is prepared, with the aid of the $\Lambda$ transformation, for calculational applications. A phenomenological theory of the degenerate Bose fluid, based on the double-quasiparticle model, is presented, but no significant physical implications are deduced from it.

A formalism is presented for the calculation of the quantum-mechanical third virial coefficient in terms of two- and three-body phase shifts. This is done for Boltzmann statistics and in the absence of bound states. The method is based on the expansion of three-body wave functions in terms of hyperspherical harmonics in six-dimensional space. Basic scattering equations are set down, including expressions for three-particle $S$, $T$, and $R$ matrices and for the three-body phase shifts. Connection is then made with statistical mechanics and an expression proposed for the evaluation of the Boltzmann three-body cluster in terms of these three-body phase shifts. Using this method, the behavior of the third virial coefficient of a gas subject to binary square-well interactions is studied in the limit when $T$ approaches zero.

A method is given for obtaining the density matrix or matrices, expressible in terms of a given set of spin geminals, which are as nearly $N$ representable as possible. A measure of $N$ representability is introduced as the norm of the antisymmetric component of a normalized wave function leading to the density matrix. An error estimate is given for the maximum deviation of an energy calculated with the density matrix from that calculated with an antisymmetric wave function. Linear variational parameters may occur in the density matrix, and their number can be increased at the expense of $N$ representability. Implementation of the method requires diagonalization of a large matrix, and the consequences of truncation are considered. The method is also related to the problem of the exact $N$ representability of a given density matrix.

The thermodynamics of critical points in multicomponent systems, more generally systems with more than two independent variables (including binary fluid mixtures, the helium $\lambda$ transition, order-disorder transitions in alloys, and antiferromagnetism) are discussed from a unified geometrical point of view, in analogy with one-component (liquid-vapor and simple-ferromagnetic) systems. It is shown that, from a few simple postulates, the qualitative behavior near the critical point of quantities such as compressibilities, susceptibilities, and heat capacities, with different choices of the variables held fixed, can be easily predicted. A number of seemingly exceptional cases (such as critical azeotropy), which arise when critical or coexistence surfaces bear an "accidental" geometrical relationship with the thermodynamic coordinate axes, are explained in terms of the same postulates. The predicted results are compared with several theoretical models and experimental data for a variety of systems.

The propagation of electromagnetic radiation through a resonant medium is studied. The slowly varying envelope of the electromagnetic field is assumed to have a slowly varying phase in addition to a slowly varying amplitude. The nonlinear coupled equations of motion are solved exactly under rather general assumptions. Three classes of solutions are obtained; two of them reduce to the solutions of Crisp and of McCall and Hahn when the slowly varying phase is zero. Because of the gauge invariance of the equations of motion, there is a term proportional to the vector potential in addition to the terms of its space and time derivatives in the equation describing the evolution in time of the electromagnetic field. Hence, a characteristic frequency exists in the infrared region below which the wave number becomes imaginary.

The Liouville equation for the $\mathrm{XY}$ model is solved exactly, and the magnetization is computed explicitly. Nonergodic behavior of the magnetization is found for a general class of time-dependent magnetic fields.

An atomic-beam magnetic-resonance apparatus was used to measure the difference in the quadratic Stark shifts of Zeeman levels in the ground states of Na and K. The results of this work are reported. Also, previously reported related work is discussed.