Abstract
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.
Highlights
Quantum walk is a mechanical system evolved by a discrete local unitary transformation and is regarded as a quantum version of the classical random walk [1,2]
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
The mathematics of quantum walk with variable parameters is not well established and difficult to derive space-time equation for its continuum limit by purely mathematical method
Summary
Quantum walk is a mechanical system evolved by a discrete local unitary transformation and is regarded as a quantum version of the classical random walk [1,2]. More recently a certain view point was added to the relation between the quantum walk and the Time Dependent Schrödinger Equation (TDSE) [10]. In quantum walk, where probability amplitude evolves instead, this transition probability matrix is replaced by the unitary banded matrix. In general we can introduce space dependent parameters x , x ,b x in the quantum walk In these parameters, b(x) means potential term in TDSE (see Appendix A). We discuss space dependent parameters x ,b x in the TDSE-type quantum walk. Type quantum walk with space dependent x with additional potential term 2 x and compared with the analytical solution for H1.
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