arXiv subjects

V. B. Mandelzweig

Publications and source records attributed to V. B. Mandelzweig.

11 recordsLinked to original sources

arXiv2007

Photoionization accompanied by excitation at intermediate photon energies

We calculate the photoionization with excitation-to photoionization ratios for atomic helium and heliumlike ions at intermediate values of the photon energies. The final state interactions between the electrons are included in the lowest order of their Sommerfeld parameter. This enables us, in contrast to purely numerical calculations to investigate the roles of various mechanisms contributing beyond the high energy limit. The system of the two bound electrons is described by the functions obtained by the Correlation Function Hyperspherical Harmonic Method. For the case of heliumlike ions we present the high energy limits as power expansion in inverse charge of the nucleus. We analyse the role of excitation of states with the nonzero orbital momenta.

quant-ph↗

arXiv2007

Highly Accurate Analytic Presentation of Solution of the Schrödinger Equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.

math-ph↗

arXiv2006

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

Ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. Our explicit analytic results are then compared with exact numerical and also with WKB solutions and it is found that our ground state wave functions, using a range of small to large coupling constants, yield a precision of between 0.1 and 1 percent and are more accurate than WKB solutions by two to three orders of magnitude. In addition, our QLM wave functions are devoid of unphysical turning point singularities and thus allow one to make analytical estimates of how variation of the oscillator parameters affects physical systems that can be described by the quartic and pure quartic oscillators.

physics.atom-ph↗

arXiv2006

Two-electron photoionization of endohedral atoms

Using $He@C_{60}$ as an example, we demonstrate that static potential of the fullerene core essentially alters the cross section of the two-electron ionization differential in one-electron energy $dσ^{++}(ω)/dε$. We found that at high photon energy prominent oscillations appear in it due to reflection of the second, slow electron wave on the $% C_{60}$ shell, which "dies out" at relatively high $ε$ values, of about 2$÷$3 two-electron ionization potentials. The results were presented for ratios $R_{C_{60}}(ω,ε)\equiv dσ^{++}(ω,ε)/dσ^{a++}(ω,ε)$, where $dσ^{a++}(ω,ε)/dε$ is the two-electron differential photoionization cross section. We have calculated the ratio $R_{i,ful}= σ_{i} ^{++}(ω)/σ_{i}^{a++}(ω)$, that accounts for reflection of both photoelectrons by the $C_{60}$ shell. We have calculated also the value of two-electron photoionization cross section $σ^{++}(ω)$ and found that this value is close to that of an isolated $He$ atom.

physics.atom-ph↗

arXiv2005

Wave Functions of Heliumlike Systems in Limiting Regions

We find approximate analytical presentation of the solutions $Ψ(r_1, r_2, r_{12})$ of Schrödinger equation for two-electron system bound by the nucleus, in the space region $r_{1,2}=0$ and $r_{12}=0$ that are of great importance for a number of physical processes. The presentation is based on the well known behavior of $Ψ(r_1, r_2, r_{12})$ near the singulartriple coalescence point. The approximate functions are compared to the locally correct ones obtained earlier by the Correlation Function Hyperspherical Harmonic (CFHH) method for helium atom, light heliumlike ions and for the negative ion of hydrogen $H^-$. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered space region. We demonstrate, how these approximate functions simplify the calculations of the high energy ionization processes.

physics.atom-ph↗

arXiv2005

Boundary Solutions of the Two-electron Schrodinger Equation at Two-particle Coalescences of the Atomic Systems

The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate CFHHM wave functions. The asymptotic solutions of the proper two-electron Schrodinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances $R$ from the triple coalescence point were presented as the second order expansion on R and lnR. The vanishing of the Fock's logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters.

physics.atom-ph↗

arXiv2005

Calculation of the photoionization with de-excitation cross sections of He and helium-like ions

We discuss the results of the calculation of the photoionization with de-excitation of excited He and helium-like ions Li$^{+}$ and B$^{3+}$ at high but non-relativistic photon energies $ω$. Several lower $^{1}S$ and $^{3}S$ states are considered. We present and analyze the ratios $R_{d}^{+\ast}$ of the cross sections of photoionization with de-excitation, $σ_{(d)}^{+\ast}(ω)$, and of the photo-ionization with excitation, $σ^{+\ast}(ω)$. The dependence of $R_{d}^{+\ast}$ on the excitation of the target object and the charge of its nucleus is presented. Apart to theoretical interest, results obtained can be verified using such long living excited state as $2^{3}S$ of He.

physics.atom-ph↗

arXiv2004

Quasilinearization Method and WKB

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers of h, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schroedinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.

math-ph↗

arXiv2004

Quasilinearization Method and Summation of the WKB Series

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the $p$-th QLM iterate in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first $2^p$ terms reproduced exactly. The QLM quantization condition leads to exact energies for the Pöschl-Teller, Hulthen, Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb Dirac equation has an accuracy of 20 significant figures.

math-ph↗

arXiv2001

Quasilinearization approach to quantum mechanics

The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precison of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.

physics.comp-ph↗

arXiv2001

Quasilinearization Approach to Nonlinear Problems in Physics with Application to Nonlinear ODEs

The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.

physics.comp-ph↗