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

Shinji Hamada, Masayuki Kawahata, Hideo Sekino

Toyohashi University of Technology, Toyohashi, Japan.

DOI: 10.4236/jqis.2013.33015                

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.

Keywords

Quantum Walk; Quantum Cellular Automaton; Time Dependent Schrödinger Equation; Advection Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Y. Aharonov, L. Davidovich and N. Zagury, “Quantum Random Walks,” Physical Review A, Vol. 48, No. 2, 1993, pp. 1687-1690. doi:10.1103/PhysRevA.48.1687
[2]	N. Konno, “Mathematics of Quantum Walk,” SangyoTosho, 2008.
[3]	P. L. Knight, E. Roldán and J. E. Sipe, “Quantum Walk on the Line as Interference Phenomenon,” Physical Review A, Vol. 68, No. 2, 2003, Article ID: 020301(R). doi:10.1103/PhysRevA.68.020301
[4]	David A. Meyer, “Quantum Mechanics of Lattice gas Automation: One Particle Plane Waves and Potential,” Physical Review E, Vol. 55, No. 5, 1997, pp. 5261-5269. doi:10.1103/PhysRevE.55.5261
[5]	B. M. Boghosian and W. Taylor, “Quantum Lattice-Gas Model for the Many-Particle Schrodinger Equation in d Dimensions,” Physical Review E, Vol. 57, No. 1, 1998, pp. 54-66. doi:10.1103/PhysRevE.57.54
[6]	F. W. Strauch, “Relativistic Quantum Walks,” Physical Review A, Vol. 73, No. 6, 2006, Article ID: 069908. doi:10.1103/PhysRevA.73.069908
[7]	A. J. Bracken, D. Ellinas and I. Smyrnakis, “Free-Diracparticle Evolution as a Quantum Random Walk,” Physical Review A, Vol. 75, No. 2, 2007, Article ID: 022322. doi:10.1103/PhysRevA.75.022322
[8]	C. M. Chandrashekar, S. Banerjee andR. Srikanth, “Relationship between Quantum Walks and Relativistic Quantum Mechanics,” Physical Review A, Vol. 81, No. 6, 2010, Article ID: 062340. doi:10.1103/PhysRevA.81.062340
[9]	A. Ahibrecht, H. Vogts, A. H. Werner, and R. F. Werner, “Asymptotic Evolution of Quantum Walks with Random Coin,” Journal of Mathematical Physics, Vol. 52, No. 4, 2011, Article ID: 042201. doi:10.1063/1.3575568
[10]	H. Sekino, M. Kawahata and S. Hamada, “A Solution of Time Dependent Schrodinger Equation by Quantum Walk,” Journal of Physics Conference Series (JPCS), Vol. 352, No. , 2012, Article ID: 012013. doi:10.1088/1742-6596/352/1/012013
[11]	I. Daubechies, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992. doi:10.1137/1.9781611970104
[12]	R. P. Feynman and A. R. Hibbs, “Quantum Mechanics and Path Integrals,” McGraw Hill, 1965.
[13]	R. Courant, K. Friedrichs and H. Lewy, “On the Partial Difference Equations of Mathematical Physics,” IBM Journal of Research and Development, Vol. 11, No. 2, 1967, pp. 215-234.
[14]	P. Høyer, “Efficient Quantum Transforms,” Unpublished, 1997. http://arxiv.org/abs/quant-ph/9702028v1