Plasma Confined Ground and Excited State Helium Atom: A Comparison Theorem Study Using Variational Monte Carlo and Lagrange Mesh Method

The energy eigenvalues of the ground state helium atom and lowest two excited states corresponding to the configurations 1s2s embedded in the plasma environment using Hulthén, Debye–Hückel and exponential cosine screened Coulomb model potentials are investigated within the variational Monte Carlo method, starting with the ultracompact trial wave functions in the form of generalized Hylleraas–Kinoshita functions and Guevara–Harris–Turbiner functions. The Lagrange mesh method calculations of energy are reported for the He atom in the ground and excited 1S and 3S states, which are in excellent agreement with the variational Monte Carlo results. Interesting relative ordering of eigenvalues are reported corresponding to the different screened Coulomb potentials in the He ground and excited electronic states, which are rationalized in terms of the comparison theorem of quantum mechanics.

Hydrogen properties in an organic molecule revealed by XFEL and electron crystallography

Structure analysis of small crystals is important in synthetic organic chemistry, pharmaceutical and material sciences, and related areas, as the conformation of these molecules may differ in large and small crystals, thus affecting the interpretation of their functional properties and drug efficacy. From small crystals, X-ray and electron beams could furnish electron densities and Coulomb potentials of target molecules, respectively. The two beams provide distinctly different information, and this potential has not been fully explored. Here we present the detailed structure of an organic molecule, rhodamine-6g by X-ray free-electron laser (XFEL) and electron crystallography from the same sample batch of microcrystals. This is the first organic molecular structure determined using XFEL at subatomic resolution. Direct comparison between the electron-density and the Coulomb-potential maps together with theoretical models based on Poisson’s equation shows that the position of hydrogen atoms depends on bond type and charge distribution. The combined approach could lead to better insights into their chemical and/or binding properties for a broad range of organic molecules.

Superintegrability of Kontsevich matrix model

Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $$\langle character\rangle \,\sim character$$ ⟨ c h a r a c t e r ⟩ ∼ c h a r a c t e r . This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.

Asymptotic behaviors of the semigroup of the linearized Landau operator for the very soft and Coulomb potentials

We study the asymptotic behaviors of the semigroup generated by the linearized Landau operator in the case of the very soft potentials and Coulomb potential. Compared with the hard potentials, Maxwellian molecules and moderately soft potentials, there is no spectral gap for the linearized Landau operator with the very soft and Coulomb potentials. By introducing a new decomposition of the linear Landau collision operator $L$ including an accretive operator and a relatively compact operator, we establish the complete spectrum structure for the linearized Landau operator with the very soft and Coulomb potentials and furthermore derive the time decay estimates of the corresponding semigroup in a weighted velocity space.

Gravitational and Coulomb Potentials Interference in Heliosphere

Interference of gravitational and Coulomb potentials in the entire heliosphere is considered, it is being manifested in generation of two opposite flows of charged particles: 1) that are neutral or with a small charge to the Sun, and 2) in the form of a solar wind from the Sun. According to the Einstein --- Smoluchowski relation Te(R) = eDe / µe ~ (E/N)0.75 based on the N experimental values (heavy particles number density --- the ne electron concentration), the Te electron temperature in the entire heliosphere was for the first time analytically calculated depending on the charge of the Sun and distance to it R. Calculated values of the registered ion parameters in the solar wind were compared with experimental observations. Reasons for generating the ring current in inhomogeneous heliosphere and inapplicability of the Debye theory in describing processes in the solar wind (plasma with current) are considered

Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

Path integral for quantum dynamics with position-dependent mass within the displacement operator approach

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.