# Showing papers in "Journal of Mathematical Physics in 1969"

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TL;DR: The theory of explicitly time-dependent invariants for quantum systems whose Hamiltonians are explicitly time dependent was developed in this article, where the authors derived a simple relation between eigenstates of such an invariant and solutions of the Schrodinger equation.

Abstract: The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrodinger equation. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrodinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.

1,613 citations

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TL;DR: In this paper, the equilibrium thermodynamics of a one-dimensional system of bosons with repulsive delta function interaction was derived from the solution of a simple integral equation, and the excitation spectrum at any temperature T was also found.

Abstract: The equilibrium thermodynamics of a one‐dimensional system of bosons with repulsive delta‐function interaction is shown to be derivable from the solution of a simple integral equation. The excitation spectrum at any temperature T is also found.

1,316 citations

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TL;DR: In this article, the problem of three equal particles interacting pairwise by inversecube forces (centrifugal potential) in addition to linear forces (harmonical potential) is solved in one dimension.

Abstract: The problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is solved in one dimension.

1,015 citations

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TL;DR: In this paper, the ground state properties of the Hamiltonian H = 12J ∑ i=1N σi·σi+1 + 12Jα ∑ n σI·σI+2 for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground states with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-nighbor interaction.

Abstract: Ground‐state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2 (σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest‐neighbor interactions only (α = 0) in the presence of the next‐nearest‐neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground‐state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest‐ as well as the next‐nearest‐neighbor interaction.

628 citations

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TL;DR: In this paper, the Burgers equation and Kortewegde Vries equation are derived for a wide class of nonlinear Galilean invariant systems under the weak nonlinearity and long-wavelength approximations.

Abstract: The Korteweg‐de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean‐invariant systems under the weak‐nonlinearity and long‐wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.

516 citations

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TL;DR: In this article, the problem of N quantum-mechanical equal particles interacting pairwise by inverse cube forces (''centrifugal potential'') in addition to linear forces ( ''harmonical potential''), is considered in a onedimensional space.

Abstract: The problem of N quantum‐mechanical equal particles interacting pairwise by inverse‐cube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is considered in a onedimensional space. An explicit expression for the ground‐state energy and for the corresponding wavefunction is exhibited. A class of excited states is similarly displayed.

488 citations

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TL;DR: In this paper, it was shown that, when steps can be taken to near-neighbor lattice points only, as N → ∞, the required number of steps is 〈n〉∼{N2/6,

Abstract: The following statistical problem arises in the theory of exciton trapping in photosynthetic units: Given an infinite periodic lattice of unit cells, each containing N points of which (N − 1) are chlorophyll molecules and one is a trap; if an exciton is produced with equal probability at any nontrapping point, how many steps on the average are required before the exciton reaches a trapping center for the first time? It is shown that, when steps can be taken to near‐neighbor lattice points only, as N → ∞, our required number of steps is 〈n〉∼{N2/6, linear chain,π−1NlogN, square lattice,1.5164N, single cubic lattice. The correction terms for medium and relatively small N are obtained for a number of lattices.

488 citations

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TL;DR: Using the Newman-Penrose formalism, the vacuum field equations were solved for Petrov type D in this paper, and an exhaustive set of ten metrics were obtained, including among them a new rotating solution closely related to the Ehlers-Kundt ``C'' metric.

Abstract: Using the Newman‐Penrose formalism, the vacuum field equations are solved for Petrov type D. An exhaustive set of ten metrics is obtained, including among them a new rotating solution closely related to the Ehlers‐Kundt ``C'' metric. They all possess at least two Killing vectors and depend only on a small number of arbitrary constants.

476 citations

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TL;DR: In this paper, a perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein-Gordon equation.

Abstract: A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein‐Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrodinger equation.

447 citations

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TL;DR: Algebraically degenerate solutions of the Einstein and Einstein-Maxwell equations are studied in this article, where explicit solutions are obtained which contain two arbitrary functions of a complex variable, one function being associated with the gravitational field and the other mainly with the electromagnetic field.

Abstract: Algebraically degenerate solutions of the Einstein and Einstein‐Maxwell equations are studied Explicit solutions are obtained which contain two arbitrary functions of a complex variable, one function being associated with the gravitational field and the other mainly with the electromagnetic field

373 citations

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TL;DR: In this article, a series of papers on high-frequency scattering of a scalar plane wave by a transparent sphere (square potential well or barrier) is presented, where the amplitude of the wave is characterized in terms of multiple internal reflections.

Abstract: This is Paper I of a series on high‐frequency scattering of a scalar plane wave by a transparent sphere (square potential well or barrier). It is assumed that (ka)⅓≫1,|N−1|½(ka)⅓≫1, where k is the wave‐number, a is the radius of the sphere, and N is the refractive index. By applying the modified Watson transformation, previously employed for an impenetrable sphere, the asymptotic behavior of the exact scattering amplitude in any direction is obtained, including several angular regions not treated before. The distribution of Regge poles is determined and their physical interpretation is given. The results are helpful in explaining the reason for the difference in the analytic properties of scattering amplitudes for cutoff potentials and potentials with tails. Following Debye, the scattering amplitude is expanded in a series, corresponding to a description in terms of multiple internal reflections. In Paper I, the first term of the Debye expansion, associated with direct reflection from the surface, and the second term, associated with direct transmission (without any internal reflection), are treated, both for N > 1 and for N 1, the behavior of the first term is similar to that found for an impenetrable sphere, with a forward diffraction peak, a lit (geometrical reflection) region, and a transition region where the amplitude is reduced to generalized Fock functions. For N 1 and for N < 1. In the former case, surface waves make shortcuts across the sphere, by critical refraction. In the latter one, they excite new surface waves by internal diffraction.

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TL;DR: In this article, it was shown that the Ricci tensor of an isometry group in an n-dimensional pseudo-Riemannian manifold is invertible, and that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes.

Abstract: Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n‐dimensional pseudo‐Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n ‐ p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n ‐ 2)‐parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space‐time (n = 4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source‐free Maxwell‐Einsteinequations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure‐vaccuum case).

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TL;DR: In this article, it was shown that the existence of a curvature collineation (CC) is a necessary condition for a covariant generator of field conservation laws in the theory of general relativity.

Abstract: A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.

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TL;DR: In this article, the effective dielectric constant of a broad class of heterogeneous materials (which we shall call cell materials) is determined in terms of statistical information, i.e., one point and three point correlation functions from variational principles.

Abstract: Determining the effective dielectric constant is typical of a broad class of problems that includes effective magnetic permeability, electrical and thermal conductivity, and diffusion. Bounds for these effective properties for statistically isotropic and homogeneous materials have been developed in terms of statistical information, i.e., one‐point and three‐point correlation functions, from variational principles. Aside from the one‐point correlation function, i.e., the volume fraction, this statistical information is difficult or impossible to obtain for real materials. For a broad class of heterogeneous materials (which we shall call cell materials) the functions of the three‐point correlation function that appear in the bounds of effective dielectric constant are simply a number for each phase. Furthermore, this number has a range of values 19 to ⅓ and a simple geometric significance. The number 19 implies a spherical shape, the number ⅓ a cell of platelike shape, and all other cell shapes, no matter h...

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TL;DR: In this article, the use of polar coordinates is examined in performing summation over all Feynman histories, and several relationships for the Lagrangian path integral and the Hamiltonians path integral are derived in the central force problem.

Abstract: Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central‐force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse‐square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.

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TL;DR: In this paper, the authors extended the results of Lee and Yang for spin ½ Ising ferromagnets to the case of arbitrary spin and showed that the zeros of the partition function lie on the unit circle in the complex fugacity plane.

Abstract: The following results for spin‐½ Ising ferromagnets are extended to the case of arbitrary spin: (1) the theorem of Lee and Yang, that the zeros of the partition function lie on the unit circle in the complex fugacity plane; (2) inequalities of the form ≥ , where A and B are products of spin operators; (3) the existence of spontaneous magnetization on suitable lattices. Results (2) and (3) are also extended to the infinite‐spin limit in which the spin variable is continuous on the interval −1 ≤ x ≤ 1.

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TL;DR: In this paper, the modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections, and the result is a uniform asymptotic expansion for the amplitude.

Abstract: The treatment, initiated in Paper I [J. Math. Phys. 10, 82 (1969)], of the high‐frequency scattering of a scalar plane wave by a transparent sphere is continued. The main results here are an improved theory of the rainbow and a theory of the glory. The modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections. Only the range 1

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TL;DR: In this paper, a transport equation for the intensity of electromagnetic waves by an extended underdense plasma is studied. But it is shown that this transport equation may be applied to the calculation of radar backscatter.

Abstract: Scattering of electromagnetic waves by an extended underdense plasma is studied. The analysis begins with expressions for multiple scattering of waves. An explicit account of coherent scatterings leads to modified equations. These modified equations are used to derive a transport equation for the intensity (a tensor expressed in polarization components). It is shown that this transport equation may be applied to the calculation of radar backscatter.

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TL;DR: By means of contour integrals involving arbitrary analytic functions, general solutions of the zero-rest mass field equations in flat space-time can be generated for each spin this article.

Abstract: By means of contour integrals involving arbitrary analytic functions, general solutions of the zero‐rest‐mass field equations in flat space‐time can be generated for each spin. If the contour surrounds only a simple (respectively, low‐order) pole of the function, the resulting field is null (respectively, algebraically special).

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TL;DR: In this paper, formal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.

Abstract: Formal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.

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TL;DR: The result of Nelson that the total Hamiltonian is semibounded for a self-interacting Boson field in two dimensions in a periodic box is derived by an alternate method as mentioned in this paper.

Abstract: The result of Nelson that the total Hamiltonian is semibounded for a self‐interacting Boson field in two dimensions in a periodic box is derived by an alternate method. It is more elementary in so far as functional integration is not used.

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TL;DR: In this paper, Dirac's bra and ket formalism is incorporated into a complete mathematical theory and several inadequacies are pointed out and several defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand.

Abstract: Dirac's bra and ket formalism is investigated and incorporated into a complete mathematical theory First the axiomatic foundations of quantum mechanics and von Neumann's spectral theory of observables are reviewed and several inadequacies are pointed out These defects then are remedied by extending the usual Hilbert space to a rigged Hilbert space as introduced by Gel'fand, ie, a triplet Φ⊂H⊂Φ′ where H is a Hilbert space, Φ a dense subspace of H provided with a new (finer) topology, Φ′ the dual of Φ It is shown that this mathematical structure, together with the Schwartz nuclear theorem, allows us to reproduce Dirac's formalism in a completely rigorous way, without losing its transparency; this makes the theory easier to handle The temporal evolution of the system and the wave equation are considered Finally the probabilistic interpretation and the physical aspects of the theory are discussed; Φ is identified with the set of all physically accessible states of the system, Φ′ with the set of all pos

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TL;DR: In this paper, it was shown that the most degenerate discrete series of unitary irreducible representations of U(2, 2), the so-called ladder representations, remain irreduceible when restricted to representations of the Poincare subgroup ISL (2, C), and the basis vectors of the canonical basis are calculated as functions of a lightlike 4-vector, which is formed by the simultaneous eigenvalues of the generators of the subgroup of translations.

Abstract: It is shown that the most degenerate discrete series of unitary irreducible representations of U(2, 2), the so‐called ladder representations, remain irreducible when restricted to representations of the Poincare subgroup ISL(2, C). They correspond to representations of this subgroup with mass zero and arbitrary integer or half‐integer helicity λ. The basis vectors of the canonical basis are calculated as functions of a lightlike 4‐vector, which is formed by the simultaneous eigenvalues of the generators of the subgroup of translations.

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TL;DR: In this paper, a simple but rigorous proof is given of the reciprocity relation for the relative motion of two inertial frames of reference, which is usually assumed as a postulate in the standard derivations of the Lorentz transformations.

Abstract: By using the principle of relativity, together with the customary assumptions concerning the nature of the space‐time manifold in special relativity, namely, space‐time homogeneity and isotropy of space, a simple but rigorous proof is given of the reciprocity relation for the relative motion of two inertial frames of reference, which is usually assumed as a postulate in the standard derivations of the Lorentz transformations without the principle of invariance of light velocity. A critical discussion is set forth of the question of eliminating the transformations with invariant imaginary velocity, which one unavoidably obtains together with the Lorentz transformations and the Galilean ones in adopting a procedure of this kind.

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TL;DR: In this paper, a uniform asymptotic theory of diffraction was proposed, which is uniformly valid near edges and shadow boundaries, but not at any caustics other than the edge.

Abstract: Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a high‐frequency asymptotic expansion. In problems involving screens with apertures, this asymptotic expansion fails at the edge of the screen and on shadow boundaries where the expansion has singularities. The uniform asymptotic theory presented here provides a new asymptotic solution of the diffraction problem which is uniformly valid near edges and shadow boundaries. Away from these regions the solution reduces to that of Keller's theory. However, singularities at any caustics other than the edge are not corrected.

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TL;DR: In this paper, the Wigner and Racah coefficients for the group SU(4)⊃[SU(2)× SU(2)] make it possible to perform the spin-isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one-and two-body operators in the supermultiplet scheme.

Abstract: Calculation of Wigner and Racah coefficients for the group SU(4)⊃[SU(2)×SU(2)] make it possible to perform the spin—isospin sums in the cfp (fractional parentage coefficients) expansion of the matrix elements of one‐ and two‐body operators in the Wigner supermultiplet scheme. The SU(4) coefficients needed to evaluate one‐ and two‐particle cfp's, the matrix elements of one‐body operators, and the diagonal matrix elements of two‐body operators are calculated in general algebraic form for many‐particle states characterized by the SU(4) irreducible representations [yy0], [y y − 1 0], [yy1], [y11], [y y − 1 y − 1], [y10], [yy y − 1], [y00], and [yyy], whose states are specified completely by the spin and isospin quantum numbers (y = arbitrary integer). Applications are made to the calculation of the matrix elements of the complete space‐scalar part of the Coulomb interaction and the space‐scalar part of the particle‐hole interaction for nucleons in different major oscillator shells.

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TL;DR: In this paper, the ground-state energy of a nonrelativistic quantum-mechanical system of N particles in gravitational interaction is derived, showing that gravitational forces do not saturate, the binding energy per particle increasing with N, like N2 for a Bose system, like (N4/3) for a Fermi system.

Abstract: Rigorous inequalities are derived for the ground‐state energy of a nonrelativistic quantum‐mechanical system of N particles in gravitational interaction. It is shown that gravitational forces do not saturate, the binding energy per particle increasing with N, like N2 for a Bose system, like (N4/3) for a Fermi system. As a by‐product, we obtain a generally valid Heisenberg‐like inequality for N‐fermion systems, expressing very simply the effect of the Pauli exclusion principle. These results are extended to the case of a system of oppositely charged particles which is shown to behave, with respect to gravitational forces, as a Fermi system as soon as particles with one sign of charge only are identical fermions. This explains quantitatively how and when gravitational forces finally predominate over Coulomb forces for large enough bodies (planets). A further extension to the case where relativistic effects enter only at the kinematical level permits us to derive rigorously from first principles the existence and an estimate of the Chandrasekhar mass limit, above which no collection of cold matter is stable (white dwarf stars).

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TL;DR: In this article, generalized Bose operators b which reduce by two the number of quanta of a Bose operator a are studied in the Fock space of a. All representations of the b's as normal-ordered (infinite degree) power series of the a's are found.

Abstract: Generalized Bose operators b which reduce by two the number of quanta of a Bose operator a are studied in the Fock space of a. All representations of the b's as normal‐ordered (infinite degree) power series of the a's are found. The unitary operators relating the irreducible components of b to a are also exhibited. The analogous result for b(k)'s which reduce the number of a quanta by k is given and the limit k → ∞ is discussed.

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TL;DR: In this article, an exact power-series expansion for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first−, second−, and third−neighbor pairs.

Abstract: The method of developing exact power‐series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first‐, second‐, and third‐neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first‐ and second‐neighbor spins, respectively (second‐neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first‐, second‐, and third‐neighbor pairs (third‐equivalent‐neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice...

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TL;DR: In this article, the theory of stochastic motion is formulated from a new point of view, and it is shown that the fundamental equations of the theory reduce to Schrodinger's equation for specific values of certain parameters.

Abstract: The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrodinger's equation for specific values of certain parameters. A generalized Fokker‐Planck‐Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrodinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum‐mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non‐Markoffian terms are briefly studied.