bluuebunny/crossref_metadata_2025_split huggingface dataset:
Stationary Expression for Effective Range
Authors: Ohmura Takashi 
Published: January 2005

A variation principle is presented for the effective range of s wave scattering. This principle applies both to the bound state and to the scattering state. It is proved that some of the stationary expressions for the effective range has the minimum value for the exact solution, thus the calculated value by using an approximate wave function gives an upper bound for the effective range.
A modified method is also proposed which may simplify the calculation in an actual application. The stationary expression gives, however, neither an upper bound nor a lower bound.
Integration in Functional Spaces and its Applications in Quantum Physics
Authors: Gel'fand I. M. , Yaglom A. M. 
Published: January 2005

This translation of the survey article by I. M. Gel'fand and A. M. Yaglom on the theory and applications of integration in functional spaces in problems of quantum physics was prepared because it was felt that such a review would be of interest and of use to mathematical physicists working in several different fields.
The article begins with a discussion of Wiener measure, after which the extension is made to the complex measure introduced by Feynman in his formulation of quantum mechanics, and examples are given of the use of these methods in quantum mechanics, quantum field theory, and quantum statistical physics. A comprehensive bibliography of works devoted to the theory and applications of functional integration methods is included.
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.
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.
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 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.
Unified Variational Formulation of Classical and Quantum Dynamics. II
Authors: Corson Edward M. 
Published: January 2005

In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle.
It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.
Representations of the Canonical Commutation Relations Describing a Nonrelativistic Infinite Free Bose Gas
Authors: Araki H. , Woods E. J. 
Published: January 2005

The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground-state representation (all particles in the zero-momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed.
It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.
Statistical Mechanics of Quenched Solid Solutions with Application to Magnetically Dilute Alloys
Authors: Morita Tohru 
Published: January 2005

The arrangement of atoms in solid solutions and alloys, prepared at high temperatures and cooled nonadiabatically, is not the one which is thermodynamically most stable. In establishing theories of phenomena related to the internal degrees of freedom of such a system, such as magnetism, one must be careful to account for this nonequilibrium distribution of atoms. In this paper, systems are treated with the aid of a fictitious equilibrium system. This fictitious system is constructed such that its thermal equilibrium properties are the same as the properties of the non-thermal-equilibrium system. Thus one can treat nonequilibrium systems by applying well known thermal equilibrium techniques to the fictitious system.
The method is illustrated via the example of a magnetically dilute alloy. Brout's result for a very dilute Ising system is obtained with the aid of the theory of classical fluids, without collecting diagrams. A method for applying the higher approximations developed for classical fluids to the present problem is suggested; calculations and discussions of which are retained for a forthcoming paper.
The Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis
Authors: Lomont J. S. , Moses H. E. 
Published: January 2005

The irreducible ray representations of the proper, orthochronous, inhomogeneous Lorentz group were originally given by Wigner in terms of a basis in which the energy and linear momenta are diagonal. In the present paper we show how the infinitesimal generators of the irreducible representations act on a basis in which the energy, the square of the angular momentum, the component of the angular momentum along the z axis, and the helicity (or circular polarization) are diagonal.
We consider representations corresponding to particles of nonzero mass, and any spin and of zero mass and finite spin. The continuous-spin case is to be treated in a later paper.
Scalar Invariants of a Rotational System in a Lie Algebra
Authors: Pease M. C. 
Published: January 2005

We consider matrix equations of the form dW/dz = [S, W], where S is a matrix function of z that is embedded in a given Lie algebra L-i.e., it is a curve in L. If the initial condition on W is in L, then W describes a curve in L. If the Killing form is used as a metric on L, then the behavior of the system is a pure rotation about an axis that is a function of z. A set of scalar invariants of such a system is obtained. These invariants form a set of conservation laws that the system obeys regardless of the detailed behavior of S(z).
If S describes a curve in some L1, which is a semisimple subalgebra of the algebra L in which the whole system is embedded, then we can split the initial condition into two parts, one of which is in L1 and generates a solution in L1, the other in a subspace that is orthogonal to L1 and that generates a curve that remains in the subspace. We can, then, obtain conservation laws that apply separately to the two parts.
The results have application to quantum mechanics since the density matrix obeys this type of equation. They also have application to coupled mode theory if we use, instead of the vector the corresponding power density spectrum matrix or the like.
Perturbation Theory of Microwave Interaction with Gyroelectric Plasmas
Authors: Nagelberg Elliott R. 
Published: January 2005

As a means of determining the effects of a uniform but arbitrarily directed magnetic field on cylindrical and spherical wave propagation in a cold, homogeneous plasma, we regard the magnetic field as a small perturbation. Assuming an expansion for the electric and magnetic fields in powers of the parameter iωg/ω, where ωg is the static gyrofrequency of the electron, we solve for the linear terms. This solution is carried out under the assumption that the fields are known for the limit of vanishing static magnetic field.
The first-order theory is then applied to cylindrical and spherical systems. When the approximate solution for the axially magnetized column is compared with the exact result, agreement is obtained provided that the static magnetic field is weak, as expected.
Random Walks on Lattices. II
Authors: Montroll Elliott W. , Weiss George H. 
Published: January 2005

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.
The number of distinct points visited after n steps on a k-dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face-centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.
The probability F(c) that a walker on a one-dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.
Most of the results in this paper have been derived by the method of Green's functions.
Scalar Invariants of a Rotational System in a Lie Algebra. II. Casimir Operators
Authors: Pease M. C. 
Published: January 2005

We consider matrix equations of the form dW/dz = [S, W] where S(z) is a matrix-valued function of z, embedded in a given semisimple Lie algbra, L. If the set Xi is a basis for L, we expand W as wiXi. We obtain general conditions on an n-index form pijk… such that p = (uivjwk…pijk…) is a constant if ui, vj, wk, etc., are solutions of the differential equation. It is shown that these forms are those that determine the vector invariants of the corresponding group.
We also ask what transformations will map any solution of the differential equation onto another, regardless of the detailed behavior of S(z). It is shown that the transformations that do this involve these same pijk….
With this identification, we are able to obtain procedures that can be used to calculate at least the two-index forms, pij. The existence of two-index forms, other than the Killing metric gij is shown to depend on the reducibility of the adjoint representation.
Finally, we show that these same forms generate Casimir operators—i.e., operators that commute with all elements of the algebra, and hence with the group associated with it. The Casimir operator so obtained is given explicitly by (pijk…XiXjXk …), where the set, Xi, is the basis of the dual algebra.
Some Theorems on the Unimodular Complex Degree of Optical Coherence
Authors: Mehta C. L. , Wolf E. , Balachandran A. P. 
Published: January 2005

The complex degree of coherence γ(r1, r2, t1 − t2) of a stationary optical field is defined as the normalized cross-correlation function (|γ| ≤ 1) of the light disturbances at two space-time points (r1, t1), (r2, t2), the disturbances being represented by means of Gabor's analytic signals.
In the present paper the general form of γ is examined under the condition that |γ| takes on the extreme value unity for all possible time differences τ = t1 − t2 (− ∞ &lt; τ &lt; ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherence theory.
Space-Time Theory of Elementary Particles. I. Born Reciprocity Principle
Authors: Shin Ernest E. H. 
Published: January 2005

The use of the Born reciprocity principle as a postulate applied to ``internal'' Minkowski space leads to a theory of elementary particles based on a purely ``space-time'' point of view. The reciprocity transformations of Born, (xμ, pμ) → (±pμ, ∓xμ), form a discrete subgroup of the x-p rotation group SU(2), which we associate with ``isotopic spin'' symmetry. The full dynamic symmetry of the eight-dimensional phase space with Minkowski metric is a 36-parameter group which contains the x-p rotation group SU(2) and the homogeneous Lorentz group ℒ as disjoint subgroups. This gives the symmetry of ``subquantic'' matter which has oscillator-rotator properties. Some physical aspects of the generators and possible forms of mass formulas for both integer and odd-half integer spins are discussed. Finally, it is suggested that the nonrelativistic counterpart of the present formalism may explain the symmetry of atomic nuclei.
``And when we have built an altar to the Invisible Light, we may set thereon the little lights for which our bodily vision is made.''
from ``The Rock'' by T. S. Eliot
On the Motion of Electrons Scattered from an Infinite Slab
Authors: Taylor C. D. 
Published: January 2005

A nonlinear integral equation of the Volterra type is derived to describe the motion of the electrons egressing from an infinite slab—one formed by parallel planes of infinite area. This motion depends on (1) the flux density of electrons egressing from the slab, and (2) the distribution of the forward components of the electrons' velocities upon egressing from the slab, i.e., the velocity components along the normal to the slab. It is proved that the nonlinear integral equation can be solved by a method of successive approximations and that the solution is unique. The integral equation is solved for two special cases.
The most important application of this work is in the study of electrons scattered by the high-density gamma flux from a nuclear explosion. Therefore, the treatment presented is made directly applicable to that study.
New Possibilities for a Unified Field Theory
Authors: Israel W. , Trollope R. 
Published: January 2005

The introduction of nonsymmetric gik in unified field theories of the Einstein-Schrödinger type is open to the objection, on group-theoretical grounds, that the symmetric and antisymmetric parts transform independently. This objection does not apply to the use of nonsymmetric Γikμ, since these quantities are irreducible under the ``extended group,'' consisting of the point transformations and the Einstein λ transformations.
We consider a theory based on symmetric gik and nonsymmetric Γikμ. The Lagrangian L is assumed to depend only on gik and the contracted curvature tensor Rik (this insures the λ invariance and transposition invariance of the theory). For simplicity, we suppose further that L involves Rik rationally and, at most, quadratically.
The resulting theory is able to account satisfactorily for the main feature of gravitation, electromagnetism, and their interaction. In particular, the theory yields the correct equations of motion for charged masses. The electromagnetic tensor is associated with the skew part of Rik, and the λ transformations correspond roughly to the gauge transformations of electrodynamics.
The Solution of a Transition Problem in a Superconducting Strip
Authors: Liniger Werner 
Published: January 2005

The isothermal transition of a strip of superconducting material from the superconducting state to the normal state, under the influence of a supercritical external magnetic field, is studied on the basis of the London theory of superconductivity. The problem can be formulated as a free boundary problem with parabolic differential equations which is solved mainly by numerical methods; an analytical solution in the form of series expansions is given for the early part of the transition.
It is found that, in the beginning, the transition in the strip behaves almost like the transition in a half-space. However, the two differ quite drastically as the transition nears completion. It is pointed out that to predict the total transition time for the strip by extrapolating from the analytical solution for the half-space is incorrect. The results show that such an estimate for the transition time would be much too small.