Applications of orthogonal polynomials to solving the Schrödinger equation

the article reveals possibilities of using hermite polynomials and related hermite weber functions to solve a wide range of problems in mathematical physicsN the obtained properties of the considered functions allow for constructing solutions for problems of wave dynamics the solution of the schrödinger integral equation constructed on their basis is highly accurate since in this case waves of matter can freely pass along the entire real axisL and there is no need for matching solutions on the finite interval

Separation of gases using ultra-thin porous layers of monodisperse nanoparticles

The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential “nanoparticle-molecule” is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10−8 m is studied

Helium passage through homogeneous ultrafine hydrocarbon layers

The present paper deals with the problem of helium atoms and methane molecules moving through a hydrocarbon layer of evenly distributed energy sources. A computational technique for integrating the Schrödinger equation based on formulation of two fundamental numerical solutions to the problem of waves passing through a barrier is suggested. A linear combination of these solutions defines the required wave function, while cross-linking with asymptotic boundary conditions allows determining the coefficients of transmission and particle reflection from the potential layer barrier

Helium passage through homogeneous ultrafine hydrocarbon layers

The present paper deals with the problem of helium atoms and methane molecules moving through a hydrocarbon layer of evenly distributed energy sources. A computational technique for integrating the Schrödinger equation based on formulation of two fundamental numerical solutions to the problem of waves passing through a barrier is suggested. A linear combination of these solutions defines the required wave function, while cross-linking with asymptotic boundary conditions allows determining the coefficients of transmission and particle reflection from the potential layer barrier

Light Isotope Separation through the Compound Membrane of Graphdiyne

The separation of isotopes of one substance is possible within the framework of the quantum mechanical model. The tunneling effect allows atoms and molecules to overcome the potential barrier with a nonzero probability. The membranes of two monoatomic layers enhance the differences in the components’ passage through the membrane, thereby providing a high separation degree of mixtures. The probability of overcoming the potential barrier by particles is found from the solving of the Schrödinger integral equation. Hermite polynomials are used to expand all the terms of the Schrödinger integral equation in a series to get a wave function. A two-layer graphdiyne membrane is used to separate the mixture

Separation of gases using ultra-thin porous layers of monodisperse nanoparticles

The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential “nanoparticle-molecule” is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10−8 m is studied

Separation of Gases Using Ultra-Thin Porous Layers of Monodisperse Nanoparticles

The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential “nanoparticle-molecule” is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10−8 m is studied

State of the helium atom inside a fullerene

The problem of motion of the helium atom inside the fullerene molecule at ultralow temperatures is considered. The solution of the Schrodinger equation is obtained by numerical methods using special functions. The potential energy of interaction of the fullerene particle with the helium atom is calculated by integrating the modified Lennard-Jones potential over the idealized surface of the hollow nanoparticle. As a result of calculations, zones of the most probable localization of the atomic particle in the states with (n, m, and k(n)) inside the C-60 fullerene were determined and visualized

Separation of Gases Using Ultra-Thin Porous Layers of Monodisperse Nanoparticles

The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential “nanoparticle-molecule” is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10−8 m is studied