Quantized Hamilton Dynamics

The paper describes the quantized Hamilton dynamics (QHD) approach that extends classical Hamiltonian dynamics and captures quantum effects, such as zero point energy, tunneling, decoherence, branching, and state-specific dynamics. The approximations are made by closures of the hierarchy of Heisenberg equations for quantum observables with the higher order observables decomposed into products of the lower order ones. The technique is applied to the vibrational energy exchange in a water molecule, the tunneling escape from a metastable state, the double-slit interference, the population transfer, dephasing and vibrational coherence transfer in a two-level system coupled to a phonon, and the scattering of a light particle off a surface phonon, where QHD is coupled to quantum mechanics in the Schrödinger representation. Generation of thermal ensembles in the extended space of QHD variables is discussed. QHD reduces to classical mechanics at the first order, closely resembles classical mechanics at the higher orders, and requires little computational effort, providing an efficient tool for treatment of the quantum effects in large systems.     

Hamilton cycles and paths in fullerenes

It has been conjectured that every fullerene, that is, every skeleton of a spherical trivalent graph whose set of faces consists of pentagons and hexagons alone, is Hamiltonian. In this article the validity of this conjecture is explored for the class of leapfrog-fullerenes. It is shown that, given an arbitrary fullerene F, the corresponding leapfrog-fullerene Le(F) contains a Hamilton cycle if the number of vertices of F is congruent to 2 modulo 4 and contains a long cycle missing out only two adjacent vertices, and thus also a Hamilton path, if the number of vertices of F is divisible by 4.