Rezumat
A formulation is proposed of the fundamentals of cumulative quantum mechanics (CQM), which allows to describe the resonant cos-waves with unlimited (with k ≠ 0), in the center of the cavity, ψn-function of electron (ψn(r) ~ cos (knr)/rk) in the hollow-space quantum resonators with any type of symmetry: plane – k = 0, spherical – k = 1 and cylinder – k = 0.5). Irregular in the center of cavity, cos-solutions are regularized of the respective type of symmetry, geometric normalization factor being equal to χ (r) = 2kπ1/2rk , with k ≠ 0 (if k = 0, then χ = 1). Stratification of the probability of finding the particle in the volume of quantum cavity similarly is determined by the energy of a particle or a full set of squares of the corresponding quantum numbers ((n-1/2)2 for the cos-waves and n2 for a sin- waves) for any type of symmetry of the quantum cavity. An analytical CQM model of polarization resonant electron capture (dynamic localization due to the self-formation of the potential barrier, cumulating this electron inside the molecule) is proposed. When the polarization capture of an electron by the allotropic forms of hollow carbon: fullerenes and nanotubes,occurs, the electron energy En > 0. The problem of polarization cumulation of the de Broglie waves of electrons is reduced to the problem of G.A. Gamow: "a quantum particle in a box, with a potential barrier on its boundary." The energy spectrum of localized states of the barrier En > 0 (metastable IQ-particle – a partially open quantum dot, line or pit), as in the case of En < 0 (FQ-stable particle – a closed quantum dot, line or pit) is determined by effective internal dimensions of a quantum box (R + rind) with polarization forces, effectively acting at a distance rind from the polarizable molecules. CQM allows for En > 0 described as a limited cumulation ψn(r)-functions for generalized interference of de Broglie-Fresnel and unlimited cumulation ψn(r)-functions to the center of the quantum cavity with the generalized interference Vysikaylo-de Broglie-Fraunhofer in hollow polarizable spherically or cylindrically symmetric quantum resonators for the de Broglie electron waves. CQM allows for the analytical calculation of eigen quantum pairs: ψn(r)-functions, respectively, the probability of finding the particle in the cavity – Wn(r) and En > 0 – eigenenergy of electrons localized in a quantum cavity (C60 and 70, etc.) by polarization. It is proved that, along with the classical energy spectrum for asymmetric ψn-functions (sin-wave) with En~n2 for cavity quantum resonators, quantum resonances for symmetric ψn-functions (cos-waves) with En ~ (n-1/2)2 can be realized.