A unified survey is presented of various theoretical approaches that have been developed to account for the novel propagation effects which may take place when extremely short pulses of coherent light interact with matter.

A detailed investigation of various aspects of the populations of interacting biological species is made, using Volterra's and other models. For a system with two species, a detailed analysis is possible, and we plot the time variation of the population of two species for a set of values of the parameters. For a many-species system, it is shown that the time average of the population $N_i$ of the $i\mathrm{th}$ species is equal to its steady state value $q_i$. For time averages of other functions of $N_i$ and $N_i{N}_j$, only the equations satisfied by them are derived. The population growth of a species is studied by assuming that the effect of other species is to introduce a random function of time in the growth equation. The resulting Fokker-Planck equation is shown to have the same form as the Schrödinger and Bloch equations. The "potential" of the Schrödinger equation depends upon the form of the number-dependent term in the growth function. For the Gompertz form, it is a simple harmonic oscillator potential and, for the Verhulst form, the Morse potential. For both the forms, the Fokker-Planck equation is explicitly solved to obtain the probability distribution $P(N, t)$ of the population as a function of time. It is shown that considerable simplification is achieved in the calculation of the steady state concentrations ($q_i\text{'}\mathrm{s}$) of the component species if Pfaffians are used. Since $q_i$ is equal to the time average of the population of the $i\mathrm{th}$ species, a great deal can be said about the stability of the population in the sense of the rarity of explosion or extinction of one or more species. If the interactions between various species and their growth coefficients (in the absence of interactions) are known, we show that a priori one can determine whether the population will be stable or not and, if not, which of the species will disappear. Also, the stability of the population is discussed when several new species are introduced. We show that the stability is dependent on how the newly introduced species interact with each other and with the population into which they are introduced. If the information about the detailed interactions between various species is absent, as is usually the case, then a statistical mechanical treatment of the population is desirable. We show that for small deviations from steady state populations, the necessary and sufficient condition for such a treatment is that the number of species is large. For arbitrary deviations the latter is a necessary condition and may not be sufficient. This statistical mechanical treatmetn provides an empirical method for calculating the interaction between two species and the stability of the population. A measure of the stability of an ecology is defined which could be used to compare the relative stabilities of two ecologies with the same macroscopic properties. The effect of changes in the interactions between varíous species, due to changes in temperature, humidity, age distribution, etc., is studied by assuming the rate constants to be random. A master equation for the probability distributions of $N_i\text{'}\mathrm{s}$ is derived. It is shown that the stationary population distribution is Poisson only if the variation in the rate constants is not too rapid. A brief outline of another stochastic model for the population growth in terms of the probabilities of birth and death of the individuals is given. Since the members of the population do not react instantaneously to any change in the environment, the prey-predator interaction does not affect the population of either prey or predator instantaneously, and the egg is not converted into an adult instantaneously, the effects of the time lags in the above processes on the behavior, in particular the stability, of the population are studied. Further generalizations of the Volterra model are discussed, and a brief review of other systems of interacting species, e.g., systems of biochemical oscillators, nervous systems, multimode lasers, systems of simultaneously growing bacteria, etc., is given. Finally, a sampling of the experiments which throw some light on the validity of Volterra's model and the statistical mechanical treatment is given.

The graphical representation of angular momentum is used as the basis of a procedure for the complete evaluation of the matrix element of a Coulomb or multipole interaction operator between atomic states having any number of open shells. The method is presented in the form of a step-by-step procedure and is designed to permit straightforward extension to the evaluation of the matrix elements of other types of tensor operators and of sums of products of Coulomb matrix elements, such as occur in the perturbation theory of configuration interaction.

We review recent work on narrow resonance models. We take the point of view that such models play a role similar to that of the Lee model in quantum field theory, and that they cannot therefore be directly compared with experiment. Examples of various aspects of these models, including general self-consistency and the construction of amplitudes with external currents are reviewed, and the related diseases are listed. A critical discussion of narrow resonance phenomenology is given. Associated questions which seem to us suitable for further study are discussed and summarized.

This article attempts to present a comprehensive review of the literature dealing with singular potentials through March 1969. The treatment is confined principally to nonrelativistic quantum mechanics, i.e., to solutions of the partial-wave Schrödinger equation with a singular potential. Some general physical and mathematical properties are given. Exact solutions are presented for those potentials for which they are available. Techniques which have been used in obtaining approximate solutions are outlined. Formal and physical applications of singular potentials are presented. Formal applications are those for which singular potentials have served as mathematical models illustrating concepts in elementary particle physics. These include applications to the Regge pole formalism, quantum field theories, and the peratization approximation. Physical applications entail those which have been made to molecular physics and to high-energy phenomenology.

This review gives a survey of the following topics: (i) results on analyticity and asymptotic growth that follow from axiomatic quantum field theory, (ii) bounds and inequalities involving cross sections at high energies, (iii) theorems and relations between cross sections requiring special assumptions, and (iv) a discussion of the Pomeranchuk theorem and of experimental and theoretical results that would hold if it were violated.

The Landau theory for infinite Fermi systems is discussed, and it is then shown how one can apply the Landau technique of functional differentiation of the energy with respect to quasiparticle occupation numbers to obtain the effective interaction, beginning from the Brueckner-Bethe approximation to the energy. Calculations of this effective interaction by Bäckman from a nucleon-nucleon potential are discussed and criticized. Results of Bäckman's calculation are compared with the parameters in Migdal's effective force, evaluated at the center of the nucleus, the latter parameters having been derived phenomenologically by fitting various nuclear phenomena. There is a large discrepancy between the calculated value of the compressibility of nuclear matter and that obtained phenomenologically. The advantages of introducing a model space in order to improve the accuracy of the calculated results is discussed, and it is pointed out that such an introduction would also be advantageous in recent calculations by Barrett and Kirson of third-order terms in the effective interaction; also, that a model space is effectively introduced in recent calculations with the Midgal theory when it comes to deriving the quadrupole-quadrupole interaction. Calculation of the effective force between two atoms in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is discussed briefly, and it is pointed out that this is a much more ambitious task than in nuclear matter. Calculations carried out within the Brueckner-Bethe formalism are reviewed.

The study of dispersion relations for external modes in molecular and complex ionic crystals is reviewed. The concept of external modes is introduced, so that the formal theory may be developed along the lines of standard Born-von Kármán formalism. The studies reported thus far in the literature are then surveyed against the background of the formal theory. The role of macroscopic electric fields in relation to external modes is next discussed, and the work of Kellermann is generalized to obtain Coulomb coefficients connected with external modes in complex ionic crystals. Finally, the group-theoretical aspects are considered, and the recent work of Maradudin and Vosko is extended to cover the external modes. A detailed illustrative example is also provided.

Within the Wightman approach to quantum field theory, we review and clarify the properties of formal charges, defined as space integrals for the fourth component of a local current. The conditions for a formal charge to determine an operator (generator) are discussed, in connection with the well-known theorems of Goldstone and of Coleman. The symmetry transformations generated by this operator—given its existence—are also studied in some detail. For generators in a scattering theory, we prove their additivity and thus provide a simple way to characterize them from their matrix elements between one-particle states. This characterization allows an immediate construction of the unitary operators implementing the symmetry transformations, and implies that all internal symmetry groups are necessarily compact. We also indicate how to construct interacting fields having definite internal quantum numbers. The present status of the proof of Noether's theorem and of its converse is discussed in the light of the rather delicate properties of formal charges.

The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.

The nature of the chemical bond in crystals is discussed. The general theories of L. Pauling based on thermochemical data and of C. A. Coulson based on valence bond concepts are compared with a recent spectroscopic theory. Particular emphasis is placed on binary crystals of formula ${\mathrm{A}}^N{\mathrm{B}}^{8-N}$ which includes most tetrahedrally coordinated semiconductors as well as crystals of the rocksalt (NaCl) family. A wide range of physical properties is discussed, including crystal structure, energy bands, elastic constants, ionization energies, and impurity states. The role of quantum-mechanical sum rules and spectral moments in constructing simplified models of bond and band behavior is explored. Stress is laid throughout on methods for incorporating quantum-mechanical effects into properties of chemical bonds through algebraic relations rather than through variational solutions of the wave equation.

Recent experimental work and its theoretical interpretation concerned with the density effect on the energy loss of charged particles in matter is reviewed with particular reference to electrons and muons. Recent proposals that radiative corrections should result in a significant reduction in the ionization loss at very high energies are analyzed in some detail. It is concluded that such an effect has not been substantiated experimentally, and that the predicted radiative corrections are likely to be small (∼1%).

A systematic list of definitions of some basic concepts in graph theory of application to physics is presented. An index, some illustrative theorems, and a brief bibliography are included.

A systematic list of definitions of some basic concepts in graph theory of application to physics is presented. An index, some illustrative theorems, and a brief bibliography are included.

Expressions are derived for the total energy loss and photon-production spectrum by the processes of Compton scattering, bremsstrahlung, and synchrotron radiation from highly relativistic electrons. For Compton scattering, the general case, the Thomson limit, and the extreme Klein-Nishina limit are considered. Bremsstrahlung is treated for the cases where the electron is scattered by a pure Coulomb field and by an atom. For the latter case the effects of shielding are discussed extensively. The synchrotron spectrum is derived for an electron moving in a circular orbit perpendicular to the magnetic field and also for the general case where the electron's motion is helical. The total photon-production spectrum is derived for each process when there is a power-law distribution of electron energies. The problems of the effects of the three processes on the electron distribution itself are considered. It is shown that if the electron loses a small fraction of its energy in a single occurrence of a process, the electron distribution function satisfies a continuity equation which is a differential equation in energy space. For the more general case where the electron can lose energy in discrete amounts (as in bremsstrahlung and extreme Klein-Nishina Compton losses), the electron distribution function satisfies an integro-differential equation. Some approximate solutions to this equation are derived for certain special cases.

The various optical, caloric, dielectric, elastic, and microwave investigations which have been used to study the tunneling states connected with the motion of atomic and molecular impurities in alkali halides are reviewed.

This review of the properties of leptons, mesons, and baryons is an updating of Review of Particle Properties, Particle Data Group [Rev. Mod. Phys. 41, 109 (1969)]. Data are evaluated, listed, averaged, and summarized in tables and wallet sheets. A data booklet is also available.