We propose a solution method of Time Dependent Schrödinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the "Time Dependent Schrödinger Equation with a space dependent imaginary diffusion coefficient" or "the advection equation with space dependent velocity fields". Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.
Time Dependent Schrödinger Equation (TDSE) with an initial Gaussian distribution, is solved by a discrete time/space Quantum Walk (QW) representing consecutive operations corresponding to a dot product of Pauli matrix and momentum operators. We call it as Schrödinge Walk (SW). Though an Hadamard Walk (HW) provides same dynamics of the probability distribution for delta-function-like initial distributions as that of the SW with a delta-function-like initial distribution, the former with a Gaussian initial distribution leads to a solution for advection of the probability distribution; the initial distribution splits into two distinctive distributions moving in opposite directions. Both mechanisms are analysed by investigating the evolution of the both amplitude components. Decoherence of the oscillating amplitudes in central region is found to be responsible for the splitting of the probability distribution in the HW.
Vapor phase decomposition of aromatic and hydroaromatic hydrocarbons by electric discharge in hydrogen was inves tigated in electric discharge sustained in a narrow gap between two dielectric barriers. Alternating current, 10,000 hertz, was used as a power source. Hydrogen containing a certain concentration of the organic vapor was preheated and fed into the discharge zone. The reactor temperature was maintained at about 300°C. and the reactor pressure was varied from 70 to 760 mm. Hg. Condensable decompo sition products were analyzed by use of mass spectroscopy and vapor phase chromatography. "p\ecomposition of organic compounds in an electric discharge generally involves fragmentation and polymerization induced by inelastic collision with high energy electrons. When hydrogen is added, active hydrogen atoms produced by electric discharge also participate in the decomposition reactions. The reaction mechanisms are complex involving excited molecules, free radicals, and ions. In this laboratory hydrocracking of coal, coal volatiles, and related materials by electric discharge in hydrogen was studied. It is postulated by Given ( I ) that coal ( vitrinite ) molecules contain aromatic and hydroaromatic structures and probably fused aromatic ring nuclei linked together by methylene or ethylene groups forming hydroaromatic rings. Many of the replaceable hydrogens in the structure are substituted by hydroxyl or carbonyl groups. Short alkyl groups and alicyclic rings may also be attached as side chains. In pyrolysis at 500°-600°C., dissociation of hydroxyl groups and dehydrogenation of naphthenic rings take place. These acts lead to formation of OH and H radicals which in turn help to break the linkages between aromatic nuclei, forming smaller, partly 316 Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on August 24, 2015 |
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