Volume 4, Issue 9, September 1963
Index of content:

Asymptotic Wave Vector and Nonrelativistic Perturbation Theory
View Description Hide DescriptionThe dual nature of perturbations in causing both transitions and persistent effects is investigated. A wave vector initially represented in terms of a set of unperturbed eigenvectors is subjected to a nondissipative perturbation. After a long time, it is assumed that this unperturbed wave vector evolves into an asymptotic wave vector in which both persistent and transition effects are present. These two effects are considered separately and the asymptotic wave vector is formally expressed as an expansion in terms of asymptotically stationary states. A time‐operator form of nonrelativistic perturbation theory which formally is very similar to the resolvent formalism is presented. In a manner similar to the resolvent formalism, diagonal and nondiagonal contributions to the development operator are considered separately. A further classification of the development operator into asymptotic and nonasymptotic parts is made. This latter form is used to obtain an explicit form for the asymptotic wave vector, and the asymptotically stationary states are identified.

Bases for the Irreducible Representations of the Unitary Groups and Some Applications
View Description Hide DescriptionIn this paper we show that sets of polynomials in the components of (2j + 1)‐dimensional vectors, solutions of certain invariant partial differential equations, form bases for all the irreducible representations of the unitary group U _{2j+1}. These polynomials will play, for the group U _{2j+1}, the same role that the solid spherical harmonics (themselves polynomials in the components of a three‐dimensional vector) play for the rotation group R _{3}. With the help of these polynomials we define and determine the reduced Wigner coefficients for the unitary groups, which we then use to derive the Wigner coefficients of U _{2j+1} by a factorization procedure. An ambiguity remains in the explicit expression for the Wigner coefficients as the Kronecker product of two irreducible representations of U _{2j+1} is not, in general, multiplicity‐free. We show how to eliminate this ambiguity with the help of operators that serve to characterize completely the rows of representations of unitary groups for a particular chain of subgroups. The procedure developed to determine the polynomial bases of U _{2j+1} seems, in principle, generalizable to arbitrary semisimple compact Lie groups.

Spin‐Component Analysis of Single‐Determinant Wavefunctions
View Description Hide DescriptionThe weights of spin components involved in a single‐determinant wavefunction are obtained. The behavior of the weights for a system of a large number of electrons is discussed.

Lower Bounds for Energy Levels of Molecular Systems
View Description Hide DescriptionIt is shown that the Hamiltonian H for a molecule in which the nuclei are regarded as fixed has the form H = Σ H _{α} + H′, where H′ is positive and each H _{α} has known eigenvalues and eigenvectors. A procedure is developed for the calculation of lower bounds to the eigenvalues of H. The optimization of the procedure and the existence of the eigenvalues are discussed.

An Exactly Soluble Model of a Many‐Fermion System
View Description Hide DescriptionAn exactly soluble model of a one‐dimensional many‐fermion system is discussed. The model has a fairly realistic interaction between pairs of fermions. An exact calculation of the momentum distribution in the ground state is given. It is shown that there is no discontinuity in the momentum distribution in this model at the Fermi surface, but that the momentum distribution has infinite slope there. Comparison with the results of perturbation theory for the same model is also presented, and it is shown that, for this case at least, the perturbation and exact answers behave qualitatively alike. Finally, the response of the system to external fields is also discussed.

Relativistic Kinetic Theory of a Simple Gas
View Description Hide DescriptionA consistent relativistic theory of transport processes in a simple gas is developed. The approach is the four‐dimensional geometric one due to Synge. Scalar and vector eigenfunctions of the linearized collision operator are derived when the scattering cross section is a simple separable function of scattering angle and relative velocity(Maxwellian particles). The bulk viscosity and thermal conductivity are computed explicitly for this case.

Kinetic Equations for Electrons and Phonons
View Description Hide DescriptionA method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.

The Number of Distinct Sites Visited in a Random Walk on a Lattice
View Description Hide DescriptionA general formalism is developed from which the average number of distinct sites visited in n steps by a random walker on a lattice can be calculated. The asymptotic value of this number for large n is shown to be for a one‐dimensional lattice and cn for lattices of three or more dimensions. The constant c is evaluated exactly, with the help of Watson's integrals, for the simple cubic, body‐centered cubic, and face‐centered cubic lattices. An analogy is drawn with an electrical network in which unit resistors replace all near‐neighbor bonds in a lattice, and the resistance of such a network on each of the three cubic lattices is evaluated.

Observable Properties of Large Relativistic Masses
View Description Hide DescriptionInterstellar space may be full of very dense and very faint stars(supernova remnants, very old white dwarfs, etc.), on whose surfaces the gravitational field intensity is very high. What, according to general relativity, are their observable properties, and what are the maximum gravitational effects as a matter of principle? To answer this, the exact test particle orbits (geodesics) of the Schwarzchild metric are derived, classified and described, on the basis of an exact classical model. The latter is of considerable help in making the properties of the orbits immediately evident, and represents the principal advantage of the present derivation over previous ones. Also, as radial coordinate, instead of the usual largely arbitrary r, the metric coefficient g _{00} = A ^{2}(r) is used. A ^{2} has immediate physical significance as the red shift ratio v(r)/v(∞), and its use also simplifies the formulas. The test‐particle scattering angle (generalizing the results of Darwin), and the differential scattering cross section, as well as the capture cross section of a sphere of radius A ^{2}(r) = A ^{2}(R) = A_{R} ^{2}, are calculated as a function of test‐particle energy, and presented graphically. The observed, augmented, angular diameter of a sphere is calculated in terms of A_{R} ^{2}, and some peculiar lenslike effects of masses, first discussed by Einstein, are reviewed. It is then pointed out that an important result of Curtis for the interior field, implies A_{R} ^{2} > 0.514, so that the exterior field extends at most down to A ^{2} = 0.514, and the region A ^{2} < 0.514—which would be the most pathological region of the exterior—cannot in fact exist, if Curtis' arguments are assumed valid. As a result, quasihyperbolic test‐particle orbits exist with pericentrum equal to the radius of any conceivable spherical mass, and the radius can therefore in principle always be determined by an asymptotic scattering experiment. As a further result, the maximum photon‐scattering angle can be no more than about 110° (and larger for slower particles), providing a cutoff at this point in the photon‐scattering cross section, and allowing a massive star to produce at most one secondary image of another star. As a practical matter, none of the effects discussed here seems large enough to be measurable today, with the sole well‐known exception of the red shift, which for the Curtis limiting sphere would be large enough to shift a visible spectrum into the near infrared.

Mathematical Analysis on the Effect of a Prolate Spheroidal Core in a Magnetic Dipole Field
View Description Hide DescriptionThe magnetic vector potential has been obtained for the case of a circular loop of current surrounding a material core of a prolate spheroidal shape, by solving Maxwell's equations and suitable boundary conditions. It is shown that this vector potential consists of two parts; the first part is that due to the loop alone, the second part being due to the presence of the core.

Addendum: Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs
View Description Hide DescriptionA gap in the proof of Theorem 1 of the cited paper is filled.