Extension of the uncertainty relations to the mixed states

In this paper we have extended the uncertainty relations to mixed states for some special cases of quantum oscillators, relations which are calculated for this kind of states by means of the earlier established virial and Hellmann-Feynman theorems. The lower bound of these uncertainty relations is estimated for a few exactly solvable potentials, too.     

On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law

In the recent literature concern has been raised regarding the validity of Kramers-Kronig relations for media with ultrasonic attenuation obeying a frequency power law. It is demonstrated, however, that the Kramers-Kronig dispersion relations for application to these types of media are available. The developed dispersion relations are compared with measurements on several liquids, and agreement is found to better than 1 m/s over the experimentally available bandwidth. A discussion regarding the validity of these dispersion relations, in particular how the dispersion relations relate to the so-called Paley-Wiener conditions, forms the conclusion.     

From the analytical structure of the pion form factor to finite energy relations for QCD condensates and their values through FESRs

Finite energy relations for QCD condensates of dimensions 4 and 6 are derived on the basis of the analytical structure of the pion form factor. “Standard” values of QCD condensates do not satisfy finite energy relations, but a number of recently determined values satisfy these relations. New values of QCD condensates are determined through FESRs.     

Improvement of Multichannel Amplitudes for the Pion-Pion Scattering Using the Dispersion Relations

The multichannel S- and P-wave amplitudes for the ππ scattering, constructed requiring analyticity and unitarity of the S-matrix and using the uniformization procedure, are elaborated using the dispersion relations with imposed crossing symmetry condition. The amplitudes are modified in the low-energy region to improve their consistency with experimental data and the dispersion relations. Agreement with data is achieved for both amplitudes from the threshold up to 1.8 GeV and with dispersion relations up to 1.1 GeV. Consequences of the applied modifications, e.g. changes of the S-wave lowest-pole positions, are presented.     

Influence of difference in the atomic volumes of the components on the mutual diffusion in liquid binary alloys

Expressions relating the partial diffusion coefficients, macroscopic flow velocity, and mutual diffusion coefficient are obtained. Unlike the known Darken relations, the relations obtained take into account the difference in the atomic volumes of the solution components. The influence of the volume factor is more pronounced when the component concentrations are significantly different.     

An analysis of the Wolfsberg-Helmholz relation

The Wolfsberg-Helmholz and other similar relations used for approximating the resonance integral, are discussed by means of two general consistency conditions which are shown not to be fulfilled by any of the studied relations. The values of the empirical Wolfsberg-Helmholz constant are also partially understood by the present analysis.     

Symmetry relations for the phase matrix of horizontally oriented particles

Seven symmetry relations, with a wide range of validity, are presented for the phase matrix of horizontally oriented particles. Three of these relations constitute a fundamental basis from which four others can be derived by making combinations. All seven relations can be used for many types of particles, including hexagonal plates and columns as well as spheroids, cylinders and cubes. Some applications are also pointed out.     

超球空间中广义角动量波函数

利用复变函数及高级超越函数的性质,得到了多维超球空间中广义解动量波函数的完备集,给出了多维超球空广义角动量的本征值,多维空间中平面波展开关系式及其它关系式。     

Finite-bandwidth effects on the causal prediction of ultrasonic attenuation of the power-law form

Kramers-Kronig (K-K) relations exist as a consequence of causality, placing nonlocal constraints on the relationship between dispersion and absorption. The finite-bandwidth method of applying these relations is examined where the K-K integrals are restricted to the spectrum of the experimental data. These finite-bandwidth K-K relations are known to work with resonant-type data and here are applied to dispersion data consistent with a power-law attenuation coefficient (exponent from 1 to 2). Bandwidth-restricted forms of the zero and once-subtracted K-K relations are used to determine the attenuation coefficient from phase velocity. Analytically, it is shown that these transforms produce the proper power-law form of the attenuation coefficient as a stand-alone term summed with artifacts that are dependent on the integration limits. Calculations are performed to demonstrate how these finite-bandwidth artifacts affect the K-K predictions under a variety of conditions. The predictions are studied in a local context as a function of subtraction frequency, bandwidth, and power-law exponent. The K-K predictions of the power-law exponent within various decades of the spectrum are also examined. In general, the agreement between finite-bandwidth K-K predictions and exact values grows as the power-law exponent approaches 1 and with increasing bandwidth.     

Relations between matrix elements of the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear Boltzmann collision integral for the axisymmetric velocity distribution are studied. Expansions in spherical Hermitian polynomials orthogonal to the Maxwellian weighting function are employed. It is shown that the nonlinear matrix elements of the collision operator are related to each other by simple relations, which are valid for arbitrary cross sections of particle interaction even if a preferential direction exists. The relations are derived from the invariance of the collision operator under the choice of basis functions or, more precisely, under both temperature and the mean velocity of the Maxwellian weighting function. The recurrent relations found allow one to calculate the matrix elements at large values of indices. This makes it possible to construct exact solutions to complicated kinetic problems.     

The relations between the decay constants of pseudoscalar mesons and the residues of the pomeron

The relations between the decay constants of pseudoscalar mesons and the residues of the pomeron are given. Based on these relations, it is shown that the relative scale between the couplings of the zero-mass Nambu-Goldstone bosons to the pomeron is determined by the spontaneous chiral symmetry breaking of the vacuum.     

Recursion relations and scattering amplitudes in the light-front formalism

The fragmentation functions and scattering amplitudes are investigated in the framework of light-front perturbation theory. It is demonstrated that, the factorization property of the fragmentation functions implies the recursion relations for the off-shell scattering amplitudes which are light-front analogs of the Berends–Giele relations. These recursion relations on the light-front can be solved exactly by induction and it is shown that the expressions for the off-shell light-front amplitudes are represented as a linear combinations of the on-shell amplitudes. By putting external particles on-shell we recover the scattering amplitudes previously derived in the literature.     

The improved quantization rule and the Langer modification

An improved quantization rule is used to obtain a generalized formulation of Langer modification. The relations between the improved quantization rule and the Langer modification are studied. Two typical quantum systems, hydrogen atom and harmonic oscillator, are studied to show the relations between them.     

Two-reduction of the super-KP hierarchy

Recursion relations are established for the residues of fractional powers of a two-reduced super-KP operator making use of the Baker-Akhiezer function. These show the integrability of the two-reduced even (or bosonic) flows of the super-KP hierarchy. Similar recursion relations are also proven for the residues of operators associated with the odd (or fermionic) flows of the Mulase-Rabin super-KP hierarchy. Due to the presence of a spectral parameter and its fermionic partner in the Baker-Akhiezer function, these recursion relations should be relevant to any attempt to prove or disprove a recent proposal that the integrable hierarchy underlying two-dimensional quantum supergravity is the Mulase-Rabin super-KP hierarchy.     

Differential relations of the nonlocal nonequilibrium thermodynamics of martensitic transformations

Differential relations describing the change in the degree of martensitic transformation with change in the external temperature and pressure are derived. As an illustration, the relations are applied to the analysis of the experimental situation of apparent violation of the Clausing-Clapeyron equation.     

Relativistic Linear Theory in the Absence of External Fields

The dielectric tensor for a multi-component, homogeneous, field-free relativistic plasma is derived in manifestly covariant form. From the dielectric tensor, linear dispersion relations are obtained explicitly when each component of the plasma is isotropic in its rest frame. If the components are relativistic Maxwellians, these dispersion relations are expressible in terms of the relativistic plasma dispersion function. Special attention is given to the Weible and two-stream instabilities and to the normal modes of a quiescent, hot electron gas. For the last case the dispersion relations are solved numerically and compared against computer simulation data. An appendix applies the formalism to cold plasmas.     

On the Onsager-Casimir reciprocal relations in a tilted Weyl semimetal

The Onsager-Casimir reciprocal relations are a fundamental symmetry of nonequilibrium statistical systems. Here we study an unusual chirality-dependent Hall effect in a tilted Weyl semimetal Co₃Sn₂S₂ with broken time-reversal symmetry. It is confirmed that the reciprocal relations are satisfied. Since two Berry curvature effects, an anomalous velocity and a chiral chemical potential, contribute to the observed Hall effect, the reciprocal relations suggest their intriguing connection.     

General solution of the two-dimensional intertwining relations for supercharges with hyperbolic (Lorentz) metrics

The supersymmetrical intertwining relations are the most productive part of the supersymmetrical method in two-dimensional Quantum Mechanics. Most interesting are relations with hyperbolic form of derivatives in supercharges. So far, several explicit solutions were found, and they provide nontrivial two-dimensional potentials which were further studied by means of supersymmetrical methods. In the present Letter the general solution of intertwining relations with hyperbolic structure of supercharges is obtained. The corresponding potentials are built explicitly, and it is evident that some of them were not known before now.     

Reciprocal relations based on the non-stationary Boltzmann equation

The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.     

Relation between the Values of Prime Numbers and Their Numbers

Journal of Experimental and Theoretical Physics - The Euler equation gives a set of an infinite number of relations between the values of prime numbers and their numbers. These relations are...