Circular Scale of Time as a Guide of the Schrödinger’s Perturbation Theory

The paper is a kind of a review which considers an investigation of the scale of time suggested by an application of the Schrödinger perturbation method, especially when the perturbation of a non-degenerate quantum state is examined. In fact the method was applied in numerous cases—also by Schrödinger himself—without any use of the notion of time. Simultaneously, because of the development of computers, their use in solving the perturbation problems gradually decreased. However, the point of importance in the paper became the time. We demonstrate that collisions of a quantum system with the perturbation potential can be arranged along a circular scale of time whose properties provide us precisely with the energy terms obtained by the Schrödinger perturbation theory. This validity of results is checked till the perturbation order N = 7.

Circular Scale of Time Applied in Classifying the Quantum-Mechanical Energy Terms Entering the Framework of the Schrodinger Perturbation Theory

The paper applies a one-to-one correspondence which exists between individual Schr?dinger perturbation terms and the diagrams obtained on a circular scale of time to whole sets of the Schr?dinger terms belonging to a definite perturbation order. In effect the diagram properties allowed us to derive the recurrence formulae giving the number of higher perturbative terms from the number of lower order terms. This recurrence formalism is based on a complementary property that any perturbation order N can be composed of two positive integer components Na , Nb combined into N in all possible ways. Another result concerns the degeneracy of the perturbative terms. This degeneracy is shown to be only twofold and the terms having it are easily detectable on the basis of a circular scale. An analysis of this type demonstrates that the degeneracy of the perturbative terms does not exist for very low perturbative orders. But when the perturbative order exceeds five, the number of degenerate terms predominates heavily over that of nondegenerate terms.

Circular Scale of Time and Energy of a Quantum State Calculated from the Schrodinger Perturbation Theory

The main facts about the scale of time considered as a plot of a sequence of events are submitted both to a review and a more detailed calculation. Classical progressive character of the time variable, present in the everyday life and in the modern science, too, is compared with a circular-like kind of advancement of time. This second kind of the time behaviour can be found suitable when a perturbation process of a quantum-mechanical system is examined. In fact the paper demonstrates that the complicated high-order Schrodinger perturbation energy of a non-degenerate quantum state becomes easy to approach of the basis of a circular scale. For example for the perturbation order N = 20 instead of 19! ≈ 1.216 × 1017 Feynman diagrams, the contribution of which should be derived and calculated, only less than 218 ≈ 2.621 × 105 terms belonging to N = 20 should be taken into account to the same purpose.

A novel algorithm for evaluating cement azimuthal density based on perturbation theory in horizontal well

Cement density monitoring plays a vital role in evaluating the quality of cementing projects,which is of great significance to the development of oil and gas.However,the presence of inhomogeneous cement distribution and casing eccentricity in horizontal wells often complicates the accurate evaluation of cement azimuthal density.In this regard,this paper proposes an algorithm to calculate the cement azimuthal density in horizontal wells using a multi-detector gamma-ray detection system.The spatial dynamic response functions are simulated to obtain the influence of cement density on gamma-ray counts by the perturbation theory,and the contribution of cement density in six sectors to the gamma-ray recorded by different detectors is obtained by integrating the spatial dynamic response functions.Combined with the relationship between gamma-ray counts and cement density,a multi-parameter calculation equation system is established,and the regularized Newton iteration method is employed to invert casing eccentricity and cement azimuthal density.This approach ensures the stability of the inversion process while simultaneously achieving an accuracy of 0.05 g/cm^(3) for the cement azimuthal density.This accuracy level is ten times higher compared to density accuracy calculated using calibration equations.Overall,this algorithm enhances the accuracy of cement azimuthal density evaluation,provides valuable technical support for the monitoring of cement azimuthal density in the oil and gas industry.

A new probability distribution model of turbulent irradiance based on Born perturbation theory

The subject of the PDF (Probability Density Function) of the irradiance fluctuations in a turbulent atmosphere is still unsettled.Theory reliably describes the behavior in the weak turbulence regime,but theoretical description in the strong and whole turbulence regimes are still controversial.Based on Born perturbation theory,the physical manifestations and correlations of three typical PDF models (Rice-Nakagami,exponential-Bessel and negative-exponential distribution) were theoretically analyzed.It is shown that these models can be derived by separately making circular-Gaussian,strong-turbulence and strong-turbulence-circular-Gaussian approximations in Born perturbation theory,which denies the viewpoint that the Rice-Nakagami model is only applicable in the extremely weak turbulence regime and provides theoretical arguments for choosing rational models in practical applications.In addition,a common shortcoming of the three models is that they are all approximations.A new model,called the Maclaurin-spread distribution,is proposed without any approximation except for assuming the correlation coefficient to be zero.So,it is considered that the new model can exactly reflect the Born perturbation theory.Simulated results prove the accuracy of this new model.

Review of Feynman’s Path Integral in Quantum Statistics:from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems.In this Review,the molecular Schrödinger equation will first be introduced,together with the BornOppenheimer approximation that decouples electronic and internuclear motions.Some effective semiclassical potentials,e.g.,centroid potential,which are all formulated in terms of Feynman’s path integral,will be discussed and compared.These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues.As a result,path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques.To complement these techniques,we will examine how Kleinert’s variational perturbation(KP)theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential.To enable the powerful KP theory to be practical for many-body molecular systems,we have proposed a new path-integral method:automated integrationfree path-integral(AIF-PI)method.Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method,we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions.The computational procedure of using our AIF-PI method,along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy(AMAZE),are also highlighted in this review.

The Pionic Deuterium and the Pion Tetrahedron Vacuum Polarization

A double-well potential model is proposed for the pionic deuterium that enables to calculate the energy split, the potential barrier height and estimate the pion tetrahedron edge length. We propose that pion tetrahedrons, πTd = ud~dũ, play a central role in the Yukawa interaction by enabling quark exchange reactions between protons and neutrons by tunneling through a potential barrier. A vacuum polarization Feynman diagram is proposed for the πTd having chains of fermion loops for the two valence quarks and anti-quarks connected by gluons. With a higher order vacuum polarization diagram, the d and u quark loops are interleaved and the chiral symmetry is broken dynamically. The proposed πTd vacuum polarization integral does not diverge in both the IR and UV limits and vanishes in the limit of an infinite pion tetrahedron condensate. We propose a new Delbruck scattering Feynman diagram that includes d and u quark and anti-quark interleaved loops. We further propose that conversion of gravitons to photons may occur via quark and anti-quark loops that describe the pion tetrahedrons dynamics in the vacuum and may also transfer gravitational waves.

A Thermodynamic Model for square-well Chain Fluid:Theory and Monte Carlo Simulation

A thermodynamic model for the freely jointed square-well chain fluids was developed based on the thermodynamic perturbation theory of Barker-Henderson, Zhang and Wertheim. In this derivation Zhang's expressions for square-well monomers improved from Barker-Henderson compressibility approximation were adopted as the reference fluid, and Wertheim's polymerization method was used to obtain the free energy term due to the bond connectivity. An analytic expression for the Helmholtz free energy of the square-well chain fluids was obtained. The expression without adjustable parameters leads to the thermodynamic consistent predictions of the compressibility factors, residual internal energy and constant-volume heat capacity for dimer, 4-mer, 8-mer and 16-mer square-well fluids. The results are in good agreement with the Monte Carlo simulation. To obtain the MC data of residual internal energy and the constant-volume heat capacity needed, NVT MC simulations were performed for these square-well chain fluids.

Benchmarking calculations of excitation energies and transition properties with spectroscopic accuracy of highly charged ions used for the fusion plasma and astrophysical plasma

Atomic radiative data such as excitation energies, transition wavelengths, radiative rates, and level lifetimes with high precision are the essential parameters for the abundance analysis, simulation, and diagnostics in fusion and astrophysical plasmas. In this work, we mainly focus on reviewing our two projects performed in the past decade. One is about the ions with Z■30 that are generally of astrophysical interest, and the other one is about the highly charged krypton(Z = 36)and tungsten(Z = 74) ions that are relevant in research of magnetic confinement fusion. Two different and independent methods, namely, multiconfiguration Dirac–Hartree–Fock(MCDHF) and the relativistic many-body perturbation theory(RMBPT) are usually used in our studies. As a complement/extension to our previous works for highly charged tungsten ions with open M-shell and open N-shell, we also mainly focus on presenting and discussing our complete RMBPT and MCDHF calculations for the excitation energies, wavelengths, electric dipole(E1), magnetic dipole(M1), electric quadrupole(E2), and magnetic quadrupole(M2) transition properties, and level lifetimes for the lowest 148 levels belonging to the 3l3configurations in Al-like W61+. We also summarize the uncertainties of our systematical theoretical calculations, by cross-checking/validating our datasets from our RMBPT and MCDHF calculations, and by detailed comparisons with available accurate observations and other theoretical calculations. The data are openly available in Science Data Bank at https://doi.org/10.57760/sciencedb.10569.

Low-lying electronic states of osmium monoxide OsO

The ground state of osmium monoxide(OsO) has long been controversial. In this paper, the low-lying Λ–S and ? electronic states of OsO have been comprehensively studied by the high-precision multi-reference calculations. The ground state of OsO is unexpectedly the closed-shell1Σ+state with a double bond instead of the previously reported3Φ or5Σ+state;after including the spin–orbit coupling effects, the ground state becomes3Π2. With the help of the theoretical spectroscopic constants and transition dipole moments, the emission spectra in the region of 405 nm–875 nm are assigned.Our results will facilitate the future studies of absorption and emission spectra of OsO.

Finite Time Domain Dynamics of Dirac Fields

In the present paper we consider the case of a Dirac field in a finite time domain and coupled to an external field. We decompose the field and its Hamiltonian in terms of creation and annihilation operators and path integrate it via Grassmannian variables techniques. In that way we obtain its finite time domain Green function. We use it in the perturbative study of the interaction of Dirac particles with classical electromagnetic waves.

Trajectory Tracking of Vertical Take-off and Landing Unmanned Aerial Vehicles Based on Disturbance Rejection Control

We investigate the trajectory tracking problem of vertical take-off and landing(VTOL) unmanned aerial vehicles(UAV), and propose a practical disturbance rejection control strategy. Firstly, the nonlinear error model is established completely by the modified Rodrigues parameters, while considering dynamics of the servo actuators. Then, a hierarchical control scheme is applied to design the translational and rotational controllers based on the time-scale property of each subsystem,respectively. And the linear extended state observer and auxiliary observer are used to deal with the uncertainties and saturation.At last, global stability of the closed-loop system is analyzed based on the singular perturbation theory. Simulation results show the effectiveness of the proposed control strategy.

Time-Delay Effects on Synchronization of Coupled Slow-Fast Systems

Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described on the basis of geometric singular perturbation theory. Then, the impact of time delay on the structure of the slow manifold of synchronized system is obtained by using the method of stability switch, and thus, time-delay effects on synchronization features are stated. It is shown the time delay cannot qualitatively affect the synchronization mechanism, however, it can result in the drift of the optimal coupling strength.

Success and Incoherence of Orthodox Quantum Mechanics

Orthodox quantum mechanics is a highly successful theory despite its serious conceptual flaws. It renounces realism, implies a kind of action-at-a-distance and is incompatible with determinism. Orthodox quantum mechanics states that Schr&oumldinger’s equation (a deterministic law) governs spontaneous processes while measurement processes are ruled by probability laws. It is well established that time dependent perturbation theory must be used for solving problems involving time. In order to account for spontaneous processes, this last theory makes use of laws valid only when measurements are performed. This incoherence seems absent from the literature.

A matrix-perturbation-theory-based optimal strategy for small-signal stability analysis of large-scale power grid

In this paper,a sensitivity matrix based approach is proposed to improve the minimum damping ratio.The proposed method also avoids burdensome deviation calculations of damping ratio of large-scale power grids when compared to the Small-Signal-Stability Constrained Optimal Power Flow(SSSC-OPF)approach.This is achieved using the Matrix Perturbation Theory(MPT)to deal with the 2nd order sensitivity matrices,and the establishment of an optimal corrective control model to regulate the output power of generating units to improve the minimum damping ratio of power grids.Finally,simulation results on the IEEE 9-bus,IEEE 39-bus and a China 634-bus systems show that the proposed approach can significantly reduce the burden of deviation calculation,while enhancing power system stability and ensuring calculation accuracy.

Canard Solutions in a Predator-Prey Model

The canard explosion phenomenon in a predator-prey model with Michaelis-Menten functional response is analyzed in this paper by employing the geometric singular perturbation theory. First, some turning points, such as, fold point, transcritical point, pitchfork point, canard point, are identified;then Hopf bifurcation, relaxation oscillation, together with the canard transition from Hopf bifurcation to relaxation oscillation are discussed.

Updating the Historical Perspective of the Interaction of Gravitational Field and Orbit in Sun-Planet-Moon System

Studying the two famous old problems that why the moon can move around the Sun and why the orbit of the Moon around the Earth cannot be broken off by the Sun under the condition calculating with F=GMm/R2, the attractive force of the Sun on the Moon is almost 2.2 times that of the Earth, we found that the planet and moon are unified as one single gravitational unit which results in that the Sun cannot have the force of F=GMm/R2 on the moon. The moon is moved by the gravitational unit orbiting around the Sun. It could indicate that the gravitational field of the moon is limited inside the unit and the gravitational fields of both the planet and moon are unified as one single field interacting with the Sun. The findings are further clarified by reestablishing Newton’s repulsive gravity.

The effect of forced oscillations on the kinetics of wave drift in an inhomogeneous plasma

The kinetics is analyzed of the drift of non-potential plasma waves in spatial positions and wavevectors due to plasma's spatial inhomogeneity. The analysis is based on highly informative kinetic scenarios of the drift of electromagnetic waves in a cold ionized plasma in the absence of a magnetic field(Erofeev 2015 Phys. Plasmas 22 092302) and the drift of long Langmuir waves in a cold magnetized plasma(Erofeev 2019 J. Plasma Phys. 85 905850104). It is shown that the traditional concept of the wave kinetic equation does not account for the effects of the forced plasma oscillations that are excited when the waves propagate in an inhomogeneous plasma.Terms are highlighted that account for these oscillations in the kinetic equations of the abovementioned highly informative wave drift scenarios.

Effective field theory approach to lepton number violatingτdecays

We continue our endeavor to investigate lepton number violating(LNV)processes at low energies in the framework of effective field theory(EFT).In this work we study the LNV tau decaysτ^(+)→lP_(i)^(+)P_(J)^(+),where L=e.μand P_(i,j)^(+)denote the lowest-lying charged pseudoscalarsπ^(+),K^(+).We analyze the dominant contributions in a series of EFTs from high to low energy scales,namely the standard model EFT(SMEFT),the low-energy EFT(LEFT),and the chiral perturbation theory(χPT).The decay branching ratios are expressed in terms of the Wilson coefficients of dimension-five and-seven operators in SMEFT and the hadronic low-energy constants.These Wilson coefficients involve the first and second generations of quarks and all generations of leptons;thus,they cannot be explored in low-energy processes such as nuclear neutrinoless double beta decay or LNV kaon decays.Unfortunately,the current experimental upper bounds on the branching ratios are too weak to set useful constraints on these coefficients.Alternatively,if we assume the new physics scale is larger than 1 TeV,the branching ratios are well below the current experimental bounds.We also estimate the hadronic uncertainties incurred in applyingχPT toτdecays by computing one-loop chiral logarithms and attempt to improve the convergence of chiral perturbation by employing dispersion relations in the short-distance part of the decay amplitudes.

Electronic band structure from first-principles Green’s function approach:theory and implementations

Electronic band structure is one of the most important intrinsic properties of a material,and is in particular crucial in electronic,photo-electronic and photo-catalytic applications.Kohn-Sham Density-functional theory(KS-DFT)within currently available local or semi-local approximations to the exchange-correlation energy functional is problematic for the description of electronic band structure.Many-body perturbation theory based on Green’s function(GF)provides a rigorous framework to describe excited-state properties of materials.The central ingredient of the GF-based many-body perturbation theory is the exchangecorrelation self-energy,which accounts for all nonclassical electron-electron interaction effects beyond the Hartree theory,and formally can be obtained by solving a set of complicated integro-differential equations,named Hedin’s equations.The GW approximation,in which the self-energy is simply a product of Green’s function and the screened Coulomb interaction(W),is currently the most accurate first-principles approach to describe electronic band structure properties of extended systems.Compared to KS-DFT,the computational efforts required for GW calculations are much larger.Various numerical techniques or approximations have been developed to apply GW for realistic systems.In this paper,we give an overview of the theory of first-principles Green’s function approach in the GW approximation and review the state of the art for the implementation of GW in different representations and with different treatment of the frequency dependence.It is hoped that further methodological developments will be inspired by this work so that the approach can be applied to more complicated and scientifically more interesting systems.