Cumulative quantum mechanics (CQM) Part II. Application of cumulative quantum mechanics in describing the Vysikaylo polarization quantum-size effects

Abstract

The processes of physical doping of nanostructured (meta-) materials have been investigated. Two types of interference and diffraction (in the center) in hollow quantum resonators for the de Broglie waves of electrons and two types of the Vysikaylo quantum-size effects due to the polarization capture of electrons in the cavity of quantum resonators are described on the basis of the cumulative quantum mechanics (CQM) formulated by the author. The first type of interference and, accordingly, diffraction in the resonator center corresponds to de Broglie-Fresnel interference (sin-wave with a node in the resonator center). This type is used to describe the localization (cumulation) of electrons in the atom quantum resonator with an atomic nucleus in the center. The second type of interference and, accordingly, diffraction is called Vysikaylo-de Broglie-Fraunhofer interference (diffraction with the antinode of the electron de Broglie wave in the center of a hollow resonator), wherein the Ψ _n -functions of the electron infinitely cumulate (are focused by a polarization “mirror”) towards the center of a hollow quantum spherically or cylindrically symmetric resonator (Ψ _n (r) ∼ cos(k _n r)/r ^k). It is shown that the cos-solutions, irregular in the resonator center, are regularized for any wave phenomena by the geometric coefficient. It is proved in the framework of CQM that, alongside with the classical energy spectrum for asymmetric Ψ _n -functions (sin waves (overtones)) with E _n ∼ n ², for hollow quantum resonators, there exist and are realized in experiments the quantum resonances for symmetric Ψ _n -functions (cos waves (the principal tone)) with E _n ∼ (n − 1/2)². The spectrum of energy states, localized by a barrier, with E _n > 0 (a metastable IQ-particle (a partially open quantum dot, line, or well)), as in the case of E _n < 0 (a stable FQ-particle (a close dot, line, or well)), is determined by the effective internal sizes of the box (R + r _ind) with polarization forces effectively acting at the distance of r _ind from the molecule. The comparison of the analytical results with the experimental observations proves convincingly the validity of the CQM application in describing the quantum-size effects in the case of physical doping of metamaterials. It has been demonstrated for the first time that, in nanocomposite materials, the pair of “eigenfunction Ψ _n —self-energy E _n” comprising the quantum state in the nanoworld noted by the basic quantum number n and in the mesoworld of nanocomposites by physically doped by traps is replaced by two nanoworld parameters: the nanocrystal diameter D and the relative resonance concentration of the modifier (a trap, such as C_{60, 70}) ζ_n. The self-assembly of hollow allotropic forms of carbon on resonant electrons is discussed.