Solution of the Time Dependent Schrödinger Equation and the Advection Equation via Quantum Walk with Variable Parameters

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We propose a solution method of Time Dependent Schr?dinger 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
Schr?dinger 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.

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http://arxiv.org/abs/quant-ph/9702028v1