The wave permeability of a compacted nanoparticle layer

The simplest example of the porous filtering system is a compacted material obtained by pressing spherical nanoparticles. Filtration characteristics of this material depend on mobility of molecules in the field of van der Waals forces.
А one-dimensional wave dynamic problem of the helium molecules motion through the ultrathin porous layer of compacted diamond nanoparticles is considered. These layers of matter make a potential barrier obstructing the passage of molecules. The permeability of the layer is derived by solving the Schrödinger equation.
The calculation technology for integration of the Schrödinger equation is suggested. It is based on two fundamental numerical solutions of the problem of waves passing through the barrier of potential forces. A linear combination of these solutions determinates the wave function. The square of this function is a probability of detecting molecules in a particular place. Linking this representation of wave function with asymptotic boundary conditions makes it possible to determine the coefficients of passing and reflecting of molecules from the barrier. The barrier is the energy of compacted nanoparticles.
This technology provides with results close to the analytical solution in particular cases. This fact allows to generalize the method to the case of molecular movement through the layer of nanoparticles and to determine the dependence between permeability and porosity of the layer.