bluuebunny/crossref_metadata_2025 huggingface dataset:
Exact Statistical Mechanics of a One-Dimensional System with Coulomb Forces
Authors: Lenard A.
Published: January 2005

A system consisting of an equal number of positively and negatively charged ``sheets'' is considered in thermal equilibrium, with motion restricted to one dimension. The configurational part of the partition function can be represented as a sum of terms, each a simple algebraic expression. The summation is performed with the technique of generating functions. The asymptotic form in the limit of an infinite system is obtained from the pole of the generating function closest to the origin. This pole is the solution of a certain transcendental equation for which an explicit analytic representation in terms of an infinite continued fraction is available. It is shown that this equation is identical with the characteristic equation associated with the even Mathieu functions of even order.

In the limit, when the ratio of interparticle force to pressure is small, the system behaves as an ideal gas, the deviations from this state being expandable in powers of the square root of this ratio. In the opposite limit of large ratio, the particles associate in pairs of opposite charge, thus behaving like an ideal gas of neutral ``molecules'' which have an internal vibrational degree of freedom.

The analysis may be generalized to include the effect of a constant external electric field. For a given pressure there is a critical field which can never be surpassed without disrupting equilibrium.
Construction of Symmetry-Adapted Functions in the Many-Particle Problem
Authors: Nesbet R. K.
Published: January 2005

A new method is presented for obtaining many-particle angular momentum eigenfunctions and matrix elements of an invariant Hamiltonian. The same technique can be used to construct symmetry-adapted functions for any group of operators that commute with the Hamiltonian, and to simplify the evaluation of matrix elements in the symmetry-adapted basis. Applied to an arbitrary configuration, the method produces orthonormal functions identical with those that would be obtained by Schmidt orthogonalization of the projections of the original basis functions of the configuration. Because of this relationship, matrix elements of the Hamiltonian are greatly simplified, but the functions are obtained without explicitly constructing the projection operators or their matrix representations. To illustrate the method, it is applied to the spin coupling of configurations with three, four, and five particles outside closed shells, and to the two 2D functions of the atomic configuration d3, in Russell-Saunders coupling. Tables of the coefficients needed to evaluate all independent matrix elements are obtained for these examples, and typical matrix elements are calculated.
Schrödinger Scattering Amplitude. I
Authors: Grossmann Alex , Wu Tai Tsun
Published: January 2005

The Schrödinger scattering amplitude for a fixed potential is studied as a function of the three components of the initial momentum, the three components of the final momentum, and the square root of the energy.
Theory of Multipole Radiation
Authors: Snowdon S. C.
Published: January 2005

The decomposition of the electromagnetic field into longitudinal, transverse electric, and transverse magnetic field types is examined in relation to a similar decomposition of the sources. In general, three independent scalar aspects of the current and charge densities must be specified to provide longitudinal and transverse electromagnetic fields. If, in addition, it is required that the sources be continuous and spatially localized functions, only two independent scalar aspects of the sources are needed to provide the electromagnetic fields outside the sources.
Molecular Collisions. III. Symmetric Top Molecules
Authors: Gioumousis George
Published: January 2005

The question of rotational energy transfer in molecular collisions is treated for the case of symmetric top molecules. The Schrödinger equation for this problem, which can be taken as an integral equation over nine variables, is reduced to a set of coupled one dimensional integral equations by means of expansions over the irreducible representations of the three-dimensional rotation group. Various selection rules are found, of which two, in indices (in the matrix elements of the potential) related to the quantum number k, have no counterpart in the previously developed theory for diatomic molecules.

The results are exhibited as cross sections for the excitation of rotational states, given the states before the collision. The rotational states for the symmetric top rotors are specified by the usual quantum numbers k, l, m, and the translational states by the velocity and direction of motion. The angular dependence of the cross section is treated by means of an expansion in spherical harmonics which is new in this context and which promises to be extremely useful in the application to transport processes.
Inverse Overlap Matrix for Periodic Arrays of Atoms
Authors: Gilbert T. L.
Published: January 2005

The calculation of the inverse overlap matrix for an infinite chain of single orbital atoms is reduced to the problem of calculating the roots of a polynomial of degree n constructed from the overlap integrals between n neighbors. The familiar method of diagonalizing the overlap matrix, inverting it, and then transforming back to the original representation is used. The final transformation leads to contour integrals which can be evaluated by the method of residues. This diagonalization method is shown to be useful also for finite chains of atoms with Born von Kárman boundary conditions, for chains of atoms with more than one orbital per atom, and for calculating the inverse root of the overlap matrix of single-orbital atoms when the only overlap is between nearest neighbors. Application of the method to two- and three-dimensional arrays of atoms leads to contour integrals containing branch points which cannot be reduced to simple analytic expressions. However, it is shown that an iterative procedure can be devised which permits the calculation of 2n terms of the Löwdin expansion for the inverse with only 2n matrix multiplications.
Variational Method for the Quantum Statistics of Interacting Particles
Authors: Girardeau M.
Published: January 2005

A variational method is developed for calculating the thermodynamic potential of quantum-mechanical many-body systems with pair-wise interactions. The method is based on Peierls' theorem and yields an upper bound to the thermodynamic potential density in the limit of an infinite system. Evaluation and minimization of the bound involves solution of a set of coupled nonlinear integral equations for the distribution function of elementary excitations and for functions defining a unitary transformation from bare particles to elementary excitations. Application of the theory to the BCS model of superconductivity reproduces the BCS results, and application to a degenerate imperfect Bose gas gives equations which are shown to be equivalent to those of Tolmachev and Wentzel.
Statistical Theory of the Energy Levels of Complex Systems. III
Authors: Dyson Freeman J.
Published: January 2005

A systematic method is developed for calculating the n-level correlation-function Rn (x1, ⋯,xn), defined as the probability for finding n levels at positions (x1, ⋯,xn), regardless of the positions of other levels. It is supposed that the levels of a complex system are statistically equivalent to the eigenvalues of a random symmetric unitary matrix of order N≫n, according to the general theory described in an earlier paper. The 2-level correlation-function is found to be R2(x1,x2)=1−{s(r)}2−{ ∫ r∞s(t)dt}{ds(r)/dr},s(r)=[sin(πr)/πr], r=|x1−x2|,the scale of energy being chosen so that the mean level-spacing is unity. It is shown how this result could in principle be used in order to determine the proportions of levels in two uncorrelated and superimposed series. An analytic expression for the distribution of nearest-neighbor level-spacings, discovered by Gaudin and Mehta, is rederived, and a similar expression is found for the distribution of spacings between next-nearest neighbors. An unexplained identity relates the nearest and next-nearest neighbor spacing distributions of a system invariant under time-reversal to the level-spacing distribution of a system without time-reversal invariance.
Vibrational Modes of Disordered Linear Chains
Authors: Rosenstock Herbert B. , McGill Richard E.
Published: January 2005

The kth normal mode of vibration (i.e., the atomic displacements) of a disordered one-dimensional lattice (masses and force constants completely arbitrary) with nearest-neighbor interaction has precisely k − 1 nodes. A fortiori, the same is true for ordered one-dimensional lattices with any number of atoms per unit cell. This theorem exhibits a close relationship between eigenfunctions in monatomic ordered lattices (to which its application has been known for many years) and disordered lattices; a relationship which appears surprising in view of recent demonstrations of the gross differences exhibited in the distribution of eigenvalues. It is thus suggested that some basic concepts of ordered lattice dynamics—propagation vector, phonon momentum, etc.—may retain some simple validity for disordered solids as well. Some numerical examples are given.
Construction of Potentials from the Phase Shifts at Fixed Energy
Authors: Newton Roger G.
Published: January 2005

The nonrelativistic potential energy between two spinless particles is deduced from a knowledge of all phase shifts at a given energy. A spherically symmetric potential is found always to exist, but it is not unique. In particular, for every energy, there exists at least one nonzero potential which causes the scattering cross section to be zero. The paper contains both the formal construction procedure and the necessary existence and uniqueness (or lack of it) proofs. Some general examples are included.
Foundations of Quaternion Quantum Mechanics
Authors: Finkelstein David , Jauch Josef M. , Schiminovich Samuel , Speiser David
Published: January 2005

A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The role played by the new imaginaries is studied. The principal conceptual difficulty concerns the theory of composite systems where the ordinary tensor product fails due to noncommutativity. It is shown that the natural resolution of this difficulty introduces new degrees of freedom similar to isospin and hypercharge. The problem of the Schrödinger equation, ``which i should appear?'' is studied and a generalization of Stone's theorem is used to resolve this problem.
Analytic Properties of Products of Field Operators
Authors: Streater R. F.
Published: January 2005

It is shown that any function of n−1 four-vectors which is analytic in the forward tube, the backward tube, and some neighborhood of the Jost points has a one-valued continuation into the extended tube. The theorem is applied to products of field operators between arbitrary states. It is shown that each matrix element of the product is analytic in the envelope of holomorphy of the union of the permuted extended tubes (the domain of analyticity of the most general causal vacuum expectation value). Thus the product of field operators may be regarded as having analytic properties.
Application of the Singular Wave Function Operator in Scattering Theory. II
Authors: Mapleton Robert A.
Published: January 2005

In a recent paper by this author, the singular wave function matrix of Mo/ller was used to obtain the scattering solution for the nonrelativistic Coulomb operator. In the present paper this singular wave function operator is defined differently. This definition is deduced from the relation of the characteristic form of the Green's function to the branch cut in the continuous part of the spectrum of the Green's function. As a consequence of this interpretation, the calculation difficulties of the previous paper do not occur. This method is also used to obtain the scattering solution for the attractive square-well potential.
Derivation of Admissibility Conditions for Wave Functions from General Quantum-Mechanical Principles
Authors: Pandres Dave
Published: January 2005

Admissibility conditions for wave functions are derived from general quantum-mechanical principles. The conditions apply in general curvilinear coordinates to functions that describe general N-particle systems. They restrict the behavior of wave functions so severely as to suggest that the Hilbert space of ordinary quantum mechanics cannot be enlarged unless the fundamental principles of quantum mechanics are modified. One of the conditions implies that all particles must be either fermions or bosons, i.e., that no particles obeying intermediate statistics can exist.
Superselection Principle and Pure States of n-Identical Particles
Authors: Galindo A. , Morales A. , Nuñez-Lagos R.
Published: January 2005

Wightman's assumption of commutative superselection rules is proved to be equivalent to Jauch's assumption of existence of a complete set of commuting observables. The analysis of coherence is carried out by using the Gel'fand representation theory and the general structure of the complete sets of compatible observables is given when the Hilbert space is separable. The above equivalence provides the physical grounds to incorporate both assumptions into the axiomatic foundations of quantum theory as a single ``superselection principle.'' To illustrate how this principle works, the problem of determining the physically realizable states of an assembly of n-identical particles is analyzed and solved.
Unusual Eigenvalues of the Energy Operator
Authors: Silveira Adel da
Published: January 2005

A generalization of quantum mechanics different from the theory proposed recently by Phipps is considered. Difficulties connected with the commutation relations of the angular momentum components defined with the help of the operators pi of the fundamental equations of Phipps are presented as new arguments in favor of the choice of Phipps of the Hermitian operators to be used in his generalized theory. The application of the equations proposed in the present paper to the motion of a particle in a central Coulombic field of forces is considered. In the Dirac theory the spectrum is different from the traditional one. As the Heisenberg postulate is now obeyed, this result shows that an unusual spectrum is not a consequence of the violation of the postulate.
Theory of Transport Coefficients. II Viscosity Coefficients of Quantum Gases Obeying the Boltzmann Statistics
Authors: Fujita S.
Published: January 2005

The viscosity coefficients of a weakly coupled and a dilute quantum gas, both obeying the classical statistics, are calculated using the method developed by Abe and the present author. In this method transport coefficients are calculated from exact formulas due to Kubo with the natural introduction of the irreversibility following the ideas of Van Hove, Prigogine, and their collaborators. It is shown explicitly that the low-density limit as well as the weak-coupling limit of the viscosity are those which are expected from the conventional calculation via the Boltzmann equation. The nature of these limits is closely related to the two well separated time scales—average duration of collision and average time between successive collisions.
Quantization of Fields with Infinite-Dimensional Invariance Groups. III. Generalized Schwinger-Feynman Theory
Authors: DeWitt Bryce S.
Published: January 2005

The formal methods of Schwinger and Feynman are applied to nonlinear field theories having elementary vertex functions of arbitrarily high order. In the first half of the paper, familiar theorems are rederived by noncanonical methods. Emphasis is given to purely formal aspects of the theory which may be expected to survive generalization to situations in which standard asymptotic conditions are inapplicable. Since the context in which the field nonlinearities are assumed to appear is that of a non-Abelian infinite-dimensional invariance group, detailed attention is given to the question of a group invariant measure for the Feynman functional integral. It is shown that the physically important part of the measure is not determined by the group.

The second half of the paper is devoted to the theory of the propagators and correlation functions which characterize the system when invariant variables are introduced. The existence of a c-number action functional Γ which contains a complete description of all quantum processes is proved. The second variational derivatives of this functional constitute the wave operator for the one-particle propagators (including all radiative corrections), and its higher derivatives are the renormalized vertex functions. A description of the renormalization process is easily carried out in terms of Γ. Finally, the implications which its existence has for quantum gravidynamics are discussed. Because it leads to nonlocal covariant equations for a complex metric tensor the way is open to transmutations of topology at the quantum level.
On the Gauge Covariance of Quantum Electrodynamics
Authors: Bialynicki-Birula I.
Published: January 2005

The gauge covariant formulation of quantum electrodynamics given by Zumino is further investigated. The method used is that of functional integration. This allows for a slight generalization of Zumino's results and provides a direct relation between the gauge condition in classical and quantum theories. All results are derived without any reference to the canonical formalism. The same procedure can be applied to the Yang-Mills field with or without a mass, leading in both cases to a renormalizable theory. The Landau gauge is studied in some detail and it is shown that in the perturbation expansion of the propagators no terms violating the gauge invariance appear. Finally, a new interpretation of the generalized Ward identity is proposed.
Integral Representations for Production Amplitudes in Perturbation Theory
Authors: Nakanishi Noboru
Published: January 2005

The integral representation proposed previously in the two-particle scattering is extended to the case of one-particle-production amplitudes. Five integral representations are found whose main parts consist of only six terms. Their support properties are investigated for the equal-mass case in every order of perturbation theory. It is shown that lowest-order graphs are by no means the representatives of analyticity in perturbation theory even in the equal-mass case.
Hard Core Produced by Orthogonality Constraints
Authors: Tani S. , Uhlenbrock D. A.
Published: January 2005

It is shown that the effect of a hard core as an external force can be reproduced by requirements of orthogonality alone. A fictitious orthonormal set of functions is introduced, which form a complete set in the domain occupied by the hard core. The actual wave function is constructed as the eigenfunction of a modified kinetic energy operator, and is required to be orthogonal to all members of the fictitious set. The limit of infinitely many constraints is carefully discussed. It turns out to be impossible to regard the hard core effect as equivalent to some kind of ``potential'' acting on free waves (for which the Hamiltonian would be well defined). However, the eigenfunction is well defined in the limit of infinitely many constraints, and is independent of the way in which the fictitious set is chosen.
A Brownian-Motion Model for the Eigenvalues of a Random Matrix
Authors: Dyson Freeman J.
Published: January 2005

A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motions without mutual interaction. By a suitable choice of initial conditions, the Brownian motion leads to an ensemble of random matrices which is a good statistical model for the Hamiltonian of a complex system possessing approximate conservation laws. The development with time of the Coulomb gas represents the statistical behavior of the eigenvalues of a complex system as the strength of conservation-destroying interactions is gradually increased. A ``virial theorem'' is proved for the Brownian-motion gas, and various properties of the stationary Coulomb gas are deduced as corollaries.
Selection Rules for Integrals of Bloch Functions
Authors: Zak J.
Published: January 2005

A simple selection rule for integrals of three Bloch functions is derived by using the irreducible representations of the complete space group. In the final formula only the characters of the small representations appear.
Green's Distributions and the Cauchy Problem for the Multi-Mass Klein-Gordon Operator
Authors: Bowman John Judson , Harris Joseph David
Published: January 2005

Explicit forms of the Green's functions (which are to be regarded as distributions in the sense of Schwartz) for the multi-mass Klein-Gordon operator in n-dimensional spaces are presented. The homogeneous Green's functions GN(x) and GN1(x), defined in the usual way by independent paths of integration in the k0 plane, are investigated in the neighborhood of the light cone. The parameter N indicates the total number of masses involved. The singularities on the light cone reflect the well-known difference between even- and odd-dimensional wave propagation. It is found that GN(x; odd n) contains a finite jump on the light cone as well as a linear combination of derivatives up to order ½(n − 2N − 1) of δ(x2); the singular part of GN1(x; odd n) consists of a logarithmic singularity ln (|x2|) along with a polynomial in (x2)−1 of degree ½(n − 2N − 1). For even-dimensional spaces, the singular part of both Green's functions consists of a polynomial in (x2)−1/2 of degree n − 2N + 1 vanishing outside the light cone for GN and vanishing inside the light cone for GN1. In all cases no singularities or finite jumps occur when the order 2N of the operator is greater than the number n + 1 of space-time dimensions. The general solution of the Cauchy problem is given both for the data carrying surface t = 0 and for arbitrary spacelike data surfaces.
Statistical Thermodynamics of Nonuniform Fluids
Authors: Lebowitz J. L. , Percus J. K.
Published: January 2005

We have developed a general formalism for obtaining the low-order distribution functions nq(r1, …, rq) and the thermodynamic parameters of nonuniform equilibrium systems where the nonuniformity is caused by a potential U(r). Our method consists of transforming from an initial (uniform) density n0 to the final desired density n(r) via a functional Taylor expansion. When n0 is chosen to be the density in the neighborhood of the r's we obtain nq as an expansion in the gradients of the density. The expansion parameter is essentially the ratio of the microscopic correlation length to the scale of the inhomogeneities. Our analysis is most conveniently done in the the grand ensemble formalism where the corresponding thermodynamic potential serves as the generating functional [with U(r) as the variable] for the nq. The transition from U(r) to n(r) as the relevant variable is accomplished via the direct correlation function which enters very naturally, relating the change in U at r2 due to a change in n at r1. It is thus essentially the matrix inverse of the two-particle Ursell function. The recent results of Stillinger and Buff on the thermodynamic potentials for non-uniform systems follow as a special case of our analysis without any recourse to the virial expansion. Thus, they hold also in the liquid region. In a succeeding paper we apply our analysis to obtain the asymptotic behavior of the radial distribution function in a uniform system.
The Inverse Problem in the Quantum Theory of Scattering
Authors: Faddeyev L. D. , Seckler B.
Published: January 2005

This report is a translation from the Russian of a survey article by L. D. Faddeyev, which appeared in Uspekhi Matem. Nauk., 14, 57 (1959). Our own interest in this article lies in its relevance to the inverse scattering problem—that is, the problem of determining information about a medium from which an electromagnetic wave is reflected, given a knowledge of the reflection coefficient. Similar questions concerning scattering phenomena in other branches of physics, e.g., in quantum mechanics, can be investigated by means of the same theory. We have therefore thought it worth-while to reproduce and distribute the translation. A good indication of the contents is given in the Introduction.
Relativistic Position Operator for Free Particles
Authors: Chakrabarti A.
Published: January 2005

Certain definitions and derivations for the relativistic center of mass are considered and are related to the position operator corresponding to the representation proposed in a previous paper. It is shown that the classical definition of Pryce has indeed a more direct correspondence with our representation than the Foldy-Wouthuysen representation, in spite of the fact that Pryce's particular method of symmetrization leads to the latter. In the process of derivation, the points of view of Synge, Pryce, and Shirokov are compared and the expressions for the velocity and the intrinsic angular momentum corresponding to the different definitions are given. In the Appendix, our operator is compared with that of Newton and Wigner for the case of spin ½.
Local Commutativity and the Analytic Continuation of the Wightman Function
Authors: Tomozawa Yukio
Published: January 2005

It is proved that the analytic continuation of the Wightman function, the vacuum expectation value of the product of field operators, due to local commutativity is single-valued in the union of the extended tubes which correspond to the Wightman functions obtained by permuting the order of the field operators in the product, and that the extended tubes, the union of them and the intersection of any two are simply connected.
Regge Poles in High-Energy Potential Scattering
Authors: Tani S.
Published: January 2005

High-energy potential scattering based on the Schrödinger equation has been investigated using Regge poles. When the Watson-Sommerfeld (W-S) integral is directly applied in the left half-plane, it is not useful at all in order to derive the scattering amplitude but gives a condition which governs the behavior of Regge poles. They are distributed with residues such that their contributions cancel against each other in the nonforward direction, provided that the integral at infinity is ignored. When Mandelstam's technique is used, the Legendre function of the first kind and the secant in the original W-S integral are replaced by the Legendre function of the second kind and the cosecant, respectively. This enables one to sum over partial waves more easily than with the conventional series. In high-energy potential scattering one cannot single out an individual Regge pole. The same conclusion follows when Khuri's representation is used.
Reduced Width Amplitude Distributions for the Unitary Ensemble
Authors: Ullah Nazakat
Published: January 2005

The reduced width amplitude distributions for a system not invariant under time reversal (unitary ensemble) are derived. The form of the joint distribution function is compared with that of the orthogonal ensemble. The calculation of the reduced width channel-channel correlation coefficient shows that only a positive width correlation is possible.
Energy-Dependent Boltzmann Equation in Plane Geometry
Authors: Bednarz Roman J. , Mika Janusz R.
Published: January 2005

The paper presents the general procedure of solving the energy-dependent Boltzmann equation in plane geometry. The particular solutions are found and then it is proved that the general solutions can be formed by superposition of particular solutions. As an illustration, the fully degenerate kernel is considered in detail and a solution in a closed form is obtained.
Representation Theory for Nonunitary Groups
Authors: Dimmock John O.
Published: January 2005

The representation theory for nonunitary groups is formulated following the same development used in the case of unitary groups, and the orthogonality relations for the corepresentation matrices are obtained. The general considerations of Wigner are used to show how one may obtain the irreducible corepresentations of a nonunitary group from the irreducible representations of its unitary subgroup.
Note on the Global Validity of the Baker-Hausdorff and Magnus Theorems
Authors: Wei James
Published: January 2005

The Baker-Hausdorff theorem states that for two given elements x and y in an associative algebra, the equation exey = ez has a solution z which lies in the Lie algebra generated by x and y. The Magnus continuous analog gives an exponential solution to a linear operator differential equation. Both theorems are valid globally for free Lie algebras of formal power series. For algebras that are not free, however, both theorems are locally but not globally valid. Some examples are given. Necessary and sufficient conditions for global validity are discussed. A superior representation in terms of a finite product of exponentials is also given.
Modified Langevin Equation for the Description of Brownian Motion
Authors: Andersen Hans C. , Oppenheim Irwin
Published: January 2005

A modified Langevin equation for the description of Brownian motion is shown to give results equivalent to those of the Langevin equation in some physical situations. The Smoluchowski equation for the probability density of the displacement of a Brownian particle is derived from the modified Langevin equation with the aid of assumptions weaker than those needed to derive the Smoluchowski equation from the ordinary Langevin equation. Sufficient conditions for the applicability of the modified Langevin equation to the calculation of configuration-space averages are obtained. Then the modified Langevin equation is applied to three simple systems, and the results are compared with those of the Langevin equation.
Singularities of Regge Trajectories and Asymptotes to Landau Curves
Authors: Polkinghorne J. C.
Published: January 2005

A new class of singularities associated with the trajectories and residues of particular Regge poles is investigated. It is shown that the singularities are associated with properties of asymptotes to Landau curves. One of the singularities corresponds to the singularity of the Regge amplitude discovered recently by Islam, Landshoff, and Taylor. The only singularities affecting physical asymptotic behavior correspond to diagrams which have all three Mandelstam spectral functions.
Asymptotic Behavior of Schrödinger Scattering Amplitudes
Authors: Sartori L.
Published: January 2005

The behavior of the S-wave scattering amplitude for large k is studied for potentials which vanish at infinity faster than any exponential, but are not cut off. The asymptotic behavior is very sensitive to the shape of the potential tail. If the potential decreases very rapidly, the growth of the Jost function resembles that with a cut-off potential. If V(r) decreases only slightly more rapidly than an exponential, then f(k) exhibits a very rapid growth in the vicinity of the positive imaginary axis. In this case also the zeros of f(k) become very dense and are concentrated near the positive imaginary axis.
Cluster Development in an N-Body Problem
Authors: Wu F. Y.
Published: January 2005

The procedure for generating useful cluster development in problems dealing with the Jastrow wavefunction, as proposed by Wu and Feenberg, is discussed in detail. The existence of the expansion is proved to all orders; also a simple rule is given for computing the expansion coefficients. The result can be considered as a generalization of the Ursell-Mayer formulas.
Interaction contra Classical Relativistic Hamiltonian Particle Mechanics
Authors: Currie Douglas G.
Published: January 2005

The relativistic Hamiltonian formalism is outlined and discussed for classical particles. The implications of the requirement that the coordinates of an event transform according to the Lorentz transformation law are discussed and expressed in a form, called the world-line conditions, which may be considered in the relativistic Hamiltonian formalism. It is then shown that the world-line conditions imply that there is no interaction in the relativistic Hamiltonian formalism; that is, the motion of any pair of particles described by the relativistic Hamiltonian formalism consists of straightline motion. In other words, if the events which compose the world lines of the particles transform according to the Lorentz transformation law, and the path of the particle is not a straight line, then this phenomena cannot be described in terms of a relativistic two-particle Hamiltonian formalism. The experimental basis for a determination of the transformation properties of an event is considered, and the relationship of the experiment to the applicability of such a formalism is discussed.
Integral Equations and Inequalities in the Theory of Fluids
Authors: Lebowitz J. L. , Percus J. K.
Published: January 2005

Using the method of functional Taylor expansion developed previously, an extensive set of equations is obtained for the distribution functions and Ursell functions in a classical fluid. These include in a systematic way many previously derived relations, e.g., Mayer-Montroll and Kirkwood-Salsburg equations. By terminating the Taylor expansion after a finite number of terms and retaining the remainder, we also obtain inequalities for the distribution functions and thermodynamic parameters of the fluid. For the case of positive interparticle potentials, we recover the inequalities first found by Lieb. For nonpositive potentials, new inequalities (some also obtained by Penrose) are derived. These inequalities are applied to the case of a hard-sphere fluid in three dimensions where they are compared with the results of machine computations and approximate theories. Different inequalities, not obtainable from the above considerations, and some properties of the fugacity expansions, are also derived.
Perturbation Approach to the Electrostatic Problem for Irregularly Shaped Conductors
Authors: Erma Victor A.
Published: January 2005

A simple and straightforward perturbation method for treating the electrostatic problem of a charged, irregularly shaped conductor is presented. The perturbation solution is generated starting from the zero-order solution for a charged sphere. The method consists of expanding the boundary condition in a Taylor series, which in effect transforms the boundary condition at the irregular boundary into a succession of boundary conditions to be satisfied at the surface of a sphere. The simplicity of the formalism consists further in applying, in a consistent manner, sufficient rather than necessary conditions on the successive correction potentials. First- and second-order expressions for the potential, surface charge density, and capacitance of irregularly shaped conductors, are derived explicitly, and an elementary theorem for the first-order capacitance is obtained. A perturbation expansion for the capacitance valid to all orders is presented. The application of the method is illustrated by calculating the capacitance of several irregularly shaped conductors. Possible generalizations to more complicated boundary-value problems are indicated.
Determinantal Solution of the Fredholm Equation with Green's Function Kernel
Authors: Brysk Henry
Published: January 2005

The Fredholm integral equation with a Green's function type of kernel has been transformed by Drukarev into an equivalent Volterra equation. It is now proven that the Neumann series solution of the Volterra equation yields the determinantal solution of the Fredholm equation. Thus, a Born approximation technique suffices to obtain the full Fredholm solution.
On the Development of Nonequilibrium Thermodynamics
Authors: De Groot S. R.
Published: January 2005

A general view of the history of nonequilibrium thermodynamics shows how two main lines of development have recently fused into a single branch of science. The field theoretical formulation of thermodynamics (leading to a balance equation for the entropy) constitutes the framework of a theory in which the Onsager reciprocal relations form the pièce de résistance, the fundamental importance of which is outlined in this paper.
Stochastic Liouville Equations
Authors: Kubo Ryogo
Published: January 2005

When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
Thermal Fluctuations in Nonlinear Systems
Authors: Van Kampen N. G.
Published: January 2005

The simple physical assumptions on which the familiar linear fluctuation theory is based do not carry over to the nonlinear case. In order to treat this case, one has to start from the master equation, and expand it in reciprocal powers of a suitable parameter (roughly speaking, the size of the system). The successive powers yield first the phenomenological law, next the linear fluctuation theory, and subsequently the influence of nonlinearity in successive approximations. The coefficients in the expansion are connected with each other by relations of the same type as the fluctuation-dissipation theorem in the linear theory. Whether or not these relations are sufficient to uniquely determine the coefficients from phenomenological data is not known at present.
Hydrodynamics of a Superfluid Condensate
Authors: Gross Eugene P.
Published: January 2005

The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
A Microscopic Approach to Superfluidity and Superconductivity
Authors: Martin Paul C.
Published: January 2005

This lecture describes certain properties of interacting Bose gases and superconductors which have recently been considered at Harvard. It is a brief resumé of work by the lecturer, P.C. Hohenberg, A. Fetter, R. Lange, and C. De Dominicis, which will be reported in mathematical detail elsewhere. It concerns attempts to derive from fundamental principles, several aspects of the macroscopic two-fluid model of London, Landau, and Tisza, the Landau phonon spectrum for bosons, and the structured condensate envisaged by Onsager and Feynman. The lecture describes new results which have been obtained concerning: (1) methods for calculating properties of large, but not necessarily infinite, condensed systems at finite temperatures; (2) methods for performing calculations in boson systems consistent with current conservation and sum rules; (3) results obtained by these methods for the phonon spectrum; (4) more accurate results for the less physical single-particle spectrum at finite temperatures; (5) results obtained for the macroscopically structured condensate in rotating systems which agree with the macroscopically inferred results of Feynman and Onsager; (6) results which treat depletion of the condensate as a result of interaction consistently and verify that nonetheless, the superfluid density, as operationally determined, is not correspondingly depleted; (7) methods in which the entropy plays a central role, for casting the macroscopic theory in a form which lays stress on the renormalized excitations of renormalized interactions; (8) the almost identical features of the mathematics for condensed Fermi and Bose systems; (9) a restatement, in this microscopic transcription, of the Feynman-Onsager argument for flux and vorticity quantization.
Time-Dependent Statistics of the Ising Model
Authors: Glauber Roy J.
Published: January 2005

The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N-member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time-varying magnetic field upon the model is discussed, and the frequency-dependent magnetic susceptibility is found in the weak-field limit. Some fluctuation-dissipation theorems are derived which relate the susceptibility to the Fourier transform of the time-dependent correlation function of the magnetization at equilibrium.
Approximate Solution of a Nonlinear Field Equation
Authors: Betts D. D. , Schiff H. , Strickfaden W. B.
Published: January 2005

Variational principles for obtaining the eigensolutions ψi of the equation ∇2ψ−ψ+ψ3=0 are developed. Variational solutions to the first two spherically symmetric eigenstates are obtained. Variational solutions of odd parity are also obtained.
On the Nonorthogonality of Generalized Momentum Eigenfunctions in Quantum Mechanics
Authors: Robinson Peter D.
Published: January 2005

The orthogonality requirement usually specified for the eigenfunctions φ of a quantum-mechanical operator P which has a continuous spectrum of eigenvalues λ, is ∫φ*(λ)φ(λ′) dλ=δ(λ−λ′). The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound-state wavefunction ψ in the form ∫ a(λ)φ(λ) dλ. The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion ψ=∫ a(λ)φ(λ) dλ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground-state hydrogen atom.

When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single-valued eigenfunctions.
Partial Wave Analysis of the Scattering of Charged Spinless Particles
Authors: Hetherington J. H.
Published: January 2005

The partial wave series for a relativistic charged spinless particle is not uniformly convergent and is difficult to evaluate numerically in the forward direction. The singularity in the scattering amplitude at the forward direction which leads to this nonuniform convergence is separated and given explicitly in a closed form so that the partial wave series may be accurately evaluated numerically.
Regge Poles and Branch Cuts for Potential Scattering
Authors: Challifour J. , Eden R. J.
Published: January 2005

The analytic properties of partial wave amplitudes are studied for complex energy and angular momentum. The properties of the wavefunctions are first obtained by standard methods in the theory of differential equations for general classes of potentials, and the effects of the dominant singular term in the potential near the origin are investigated. These include the appearance of branch cuts in the angular-momentum variable for potentials which are singular like z−2, and the location of Regge poles for more singular potentials. The trajectories of Regge poles are also studied with particular reference to their behavior in the angular-momentum plane as the energy tends to infinity. An example is given of a singular potential in which the trajectories move to infinity in a complex direction, contrary to the normal behavior for which they tend to negative integers. The real sections of Regge surfaces are also briefly discussed.
New Approach to Low-Energy Potential Scattering
Authors: Dashen Roger F.
Published: January 2005

The problem of low-energy potential scattering is reformulated in a manner suggested by the ``invariant imbedding'' techniques of transport theory. The differential equations thus obtained have several conceptual and computational advantages over the Schrödinger equation. Some new bounds and approximations are derived and a rigorous investigation of the Born approximation is carried out.
Moments of the Neutron Time-Energy Distribution and the Use of Random Functionals
Authors: Leibowitz Martin A.
Published: January 2005

The time moments of the neutron time-energy distribution are derived under the following conditions: infinite homogeneous moderator with no absorption, elastic scattering, and distribution of scattering angle independent of energy in center-of-mass frame. However, no assumption is made on the variation of the scattering cross section with energy. This is in contrast to previous papers in which a scattering cross section proportional to vδ (δ any real number) is assumed. The moments are derived in two ways: first, by means of Laplace transforms, and then through the use of random functionals. Random functionals have not been previously employed in neutron moderation theory and they offer certain advantages which are discussed.
Generalization of a Theorem by Goldberg and Sachs
Authors: Robinson I. , Schild A.
Published: January 2005

A connection is established between algebraic degeneracy of the Weyl tensor, the existence of a null geodesic shear-free congruence, and certain restrictions on the Ricci tensor which are weaker than the gravitational equations for empty space. The result is roughly, with some important qualifications, that any two of these conditions imply the third. These restrictions on the metric are shown to be invariant under conformal transformations.