Semiclassical method for calculating the energetic values of helium, lithium and beryllium atoms

In previous papers we have presented a wave model for conservative bound systems resulted from the equivalency between the Schrödinger and wave equations. We proved that the normal curves of the characteristic surface of the wave equation, denoted by C curves, are solutions of the Hamilton-Jacobi equation, written for the same system, and correspond to the same constants of motion as those resulting from the Schrödinger equation. In this paper we present a method for computing the energetic values of conservative bound systems which is based on the properties of the C curves. The method is applied to the 1s ² state of helium, 1s ²2s and 1s ²2p states of lithium and 1s ²2s ² state of beryllium. Our theoretical values are compared with experimental data taken from well-known books. The relative error of our method is less than 5 x 10^?3.     

On Shtelen’s Solution of the Free Linear Schrödinger Equation

Abstract

The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined, including its extension to n ≥ 3 spatial dimensions and its time decay; which is shown to be slower than that of the usual solution of this equation. These new solutions are then used to define certain mappings, F _n, on L ²(?ⁿ) and a number of their properties are studied; in particular, their global smoothing properties are considered. The differences between the behavior of F _n and that of analogous mappings constructed from usual solutions of the free Schrödinger equation are discussed.     

Padé approximation methods applied to the intermolecular force series

The H(1s)-H⁺, He-He, H(1s)-H(1s) and H(1s)-H(2s) interactions are considered as model systems for investigating the use of the Padé approximation method in summing the R ^-1 intermolecular force series. Various Padé approximants and partial sums of the R ^-1 expansions of the second-order Coulomb interaction energies are compared with the corresponding non-expanded results for each interaction. The computations are based on Unsöld's average energy approximation and on exact results for the H(1s)-H interaction. The results indicate that the Padé approximation method is a simple, useful way to remove some of the difficulties associated with the slow rate of convergence of the R ^-1 force series but that it does not alleviate the problems associated with the asymptotic divergent nature of the series. The results for the H(1s)-H⁺ interaction illustrate a possible difficulty in using Unsöld's method in the calculation of interaction energies.     

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation. By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T′* M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T′* M, H) is obtained. This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T′*M, H). Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.     

Kinetic equations and brownian motion for a solid in a stationary temperature gradient

A kinetic equation for the distribution function for a subsystem s interacting with a bath b maintained in a stationary non-isothermal state by reservoirs is derived by using a projection operator formalism and a perturbation expansion parameter λ appropriate to some brownian motion problems. Thus, when s is a heavy particle of mass m ₀ and b a lattice of light particles of mass m then λ = (m m ₀)^1/2. By means of an assumption about the decay of correlations of b variables in the field of s, the terms are classified as destruction terms which vanish to arbitrary order in λ for long enough times and collision terms which give well-defined integrals for long times. For the heavy particle in a lattice the leading collision terms, after linearization in the temperature gradient, lead to an equation equivalent to the generalized nonisothermal Fokker-Planck equation of Nicolis.     

Spline algorithms for the hartree-fock equation for the helium ground state

Spline algorithms are evaluated for the non-linear, integro-differential equation describing the Hartree-Fock approximation for the He 1s²¹S ground state. The error in the energy decreases as h^2K−2, where h is a grid parameter and K is the order of the spline. It is shown that for higher order splines, the method is fast and accurate, and contrary to the conclusion reached by Altenberger-Siczek and Gilbert, that spline methods are suitable for SCF atomic structure calculations. Accuracy and timing studies are presented as well as comparisons with other accurate procedures.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

Classical φ6-field theory in (1+1) dimensions. A model for structural phase transitions

The classical φ⁶-field theory in (1+1) dimensions, is considered as a model for the first order structural phase transitions. The equation of motion is solved exactly; and the presence of domain wall (kink) solutions at and below the transition point, in addition to the usual phonon-like oscillatory solutions, is demonstrated. The domain wall solutions are shown to be stable, and their mass and energies are calculated. Above the transition point there exists exotic unstable kink-like solutions which takes the particle from one hill top to the other of the potential. The partition function of the system is calculated exactly using the functional integral method together with the transfer matrix techniques which necessitates the determination of the eigenvalues of a Schrödinger-like equation. Thus the exact free energy is evaluated which in the low temperature limit has a phonon part and a contribution coming from the domain wall excitations. It was shown that this domain wall free energy differs from that calculated by the use of the domain wall phenomenology proposed by Krumhansl and Schrieffer. The exact solutions of the Schrödinger-like equation are also used to evaluate the displacement-displacement, intensity-intensity correlation functions and the probability distribution function. These results are compared with those obtained from the phenomenology as well as the φ⁴-field theory. A qualitative picture of the central peak observed in structural phase transitions is proposed.     

Large spin corrections in SYM : Still a linear integral equation

Anomalous dimension and higher conserved charges in the sl(2) sector of SYM for generic spin s and twist L are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as s→∞. Therefore they can be neglected at least up to the O(s⁰) order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the ‘thermodynamic’ (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power (lns)⁻ⁿ, n>0, thus extending the linearity validity.     

On the motion of a charged particle with radiation reaction

The object of this paper is to show that initial acceleration is uniquely determined by the condition that the physical solutions of the Dirac-Lorentz equation are regular at 2e ²/3mc ³=0. The method for obtaining this initial acceleration in the general case is given.     

On local symmetries of the Schrödinger equation

The multi-symplectic approach to the Schrödinger equation with a potential V = V(t,x^k) is given. The condition for a vector field X in the multi-symplectic space to be a symmetry field is found. For a spherically symmetrical potential all such symmetry fields are effectively found.The one-to-one correspondence between solutions of the free Schrödinger equation and solutions of the oscillator problem is given. This enables us to give a new geometric interpretation of the non-typical, given by A.O. Barut, symmetry of the Schrödinger equation.     

An asymptotic series solution to the Schrödinger equation for a model potential

Using the method of Frobenius, the attempt to find a closed-form solution to the Schrödinger equation for a model potential V( r) = -Ze²/?{r²+a²}V\left( r\right) =-Ze^{2}/\sqrt{r^{2}+a^{2}} is described. Inspection of the series solution in the asymptotic region suggests a substitution of a four termrecurrence relation by a two term relation in order to extract explicit solutions that approach to those obtained for the hydrogen atom as a? 0a\rightarrow 0 smoothly. The resulting approximate solutions are compared with numerical ones and shown to provide a good basis set for high Rydberg states.     

Numerical solution of a stationary nonisothermal two-phase filtration problem by the steadying method

The structure of solutions of a stationary nonisothermal problem of the two-phase filtration of immiscible fluids is studied numerically. The character of the convergence of nonstationary solutions to stationary ones is investigated. It is shown that at different parameter values the solution may have an interval, where s(x) ≡ 0 or s(x) ≡ 1. The temperature effect on the structure of the solutions of the equation for water saturation is investigated. The work was partially supported by the SB RAS (Integration Project No. 117).     

Exact and approximate solutions to Schrödinger’s equation with decatic potentials

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax ¹⁰ + bx ⁸ + cx ⁶ + dx ⁴ + ex ², a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.     

Charge overlap effects dependence on the nature of the interaction

The He-He, He-H(1s), H(1s)-H(1s) and H(1s)-H(2s) interactions are considered as model systems for studying how charge overlap effects in second-order dispersion energies vary as a function of the nature of the interacting species. The non-expanded second-order energy and the corresponding multipole R ^-1 expansions, through all powers of R ^-1, are obtained for each interaction using Unsöld's average energy approximation. The results are used to discuss the limitations of the usefulness of the R ^-1 expansions.     

Mössbauer investigation of iron uptake in wheat

Iron uptake and distribution in wheat roots were studied with ⁵⁷Fe Mössbauer spectroscopy. Plants were grown both in iron sufficient and in iron deficient nutrient solutions. Mössbauer spectra of the frozen iron sufficient roots exhibited three iron(III) components with the typical average Mössbauer parameters of δ?= 0.50 mm s^???1, Δ?= 0.43 mm s^???1, δ?= 0.50 mm s^???1, Δ?= 0.75 mm s^???1 and δ?= 0.50 mm s^???1, Δ?= 1.20 mm s^???1 at 80 K. These doublets are very similar to those obtained earlier for cucumber [0], which allows us to suppose that iron is stored in a very similar way in different plants. No ferrous iron could be identified in any case, not even in the iron deficient roots, which confirms the mechanism proposed for iron uptake in the graminaceous plants.     

Critical binding of diatomic molecules

The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α₀)², where α₀ is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k ² is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.     

Estimation of transition probabilities in a hydrogen atom under the influence of a short electric pulse based on the path integral

Schemes for estimation of the path integral on the basis of the saddle-point method are tested. Estimation of the excitation probabilities of 1s–2s, 2p transitions in a hydrogen atom under the influence of a short electric pulse is chosen as a test problem. The obtained results are compared with the previous calculations based on the solution of the nonstationary Schrödinger equation by the finite element method.     

On a systematic derivation of the quark-antiquark potential

A systematic method of deriving the quark-antiquark potential is developed. The whole procedure is organized as a 1/c expansion. Up to order 1/c², the Eichten-Feinberg spin-dependent potential is recovered and new velocity-dependent terms are obtained. As essential step of the derivation is the identification of a Pauli-type two-particle propagator, which satisfies a suitable Schrödinger equation. The short- and large-distance behaviour of the potentials is studied.     

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V _s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P _I ² equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P _I ² equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P _I ² equation pops up in the n ^−2/7-term of the asymptotics.