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A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrodinger Equation

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DOI: 10.4236/ajcm.2012.23022    4,872 Downloads   9,903 Views   Citations

ABSTRACT

The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schr?dinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

F. Moxley III, F. Zhu and W. Dai, "A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrodinger Equation," American Journal of Computational Mathematics, Vol. 2 No. 3, 2012, pp. 163-172. doi: 10.4236/ajcm.2012.23022.

References

[1] S. Brandt and H. D. Dahmen, “The Picture Book of Quantum Mechanics,” Springer Verlag, Berlin, 1995. doi:10.1007/978-1-4684-0233-9
[2] D. J. Griffiths, “Introduction to Quantum Mechanics. Englewood Cliffs,” Prentice-Hall, NJ, 1995.
[3] D. M. Sullivan, “Electromagnetic Simulation Using The FDTD Method,” IEEE Press. New York, 2000. doi:10.1109/9780470544518
[4] P. B. Visscher, “A Fast Explicit Algorithm for the Time Dependent Schrodinger Equation,” Comput. Phys., Vol. 5, 1991, pp. 596-598. doi:10.1063/1.168415
[5] A. Arnold, M. Ehrhardt and I. Sofronov, “Discrete Transparent Boundary Conditions for the Schr?dinger Equation: Fast Calculation, Approximation and Stability,” Communications in Mathematical Science, Vol. 1, 2003, pp. 501-556.
[6] A. Askar and A. S. Cakmak, “Explicit Integration Method for the Time-Dependent Schr?dinger Equation,” J Chem Phys, Vol. 68, 1978, pp. 2794-2798. doi:10.1063/1.436072
[7] W. Bao, S. Jin and P. Markowich, “On Time-Splitting Spectral Approximations for the Schr?dinger Equation in the Semiclassical Regime,” J Comput Phys, Vol. 26, 2005, pp. 487-524.
[8] N. Carjan, M. Rizea and S. Strottman, “Efficient Numerical Solution of the Time-Dependent Schr?dinger Equation for Deep Tunneling,” Romanian Reports in Physics, Vol. 55, 2003, pp. 555-579.
[9] T. F. Chan, D. Lee and L. Shen, “Stable Explicit Schemes for Equations of the Schr?dinger Type,” SIAM J Numer Anal, Vol. 23, 1986, pp. 274-281. doi:10.1137/0723019
[10] H. Chen and B. D. Shizgal, “The Quadrature Discretization Method in the Solution of the Schr?dinger Equation,” J Chem, Vol. 24, 1998, pp. 321-343.
[11] Z. Chen, J. Zhang and Z. Yu, “Solution of the Time-Dependent Schr?dinger Equation with Absorbing Boundary Conditions,” Journal of Semiconductors, Vol. 30, 2009, pp. 012001. doi:10.1088/1674-4926/30/1/012001
[12] W. Dai, “An Unconditionally Stable Three-Level Explicit Difference Scheme for the Schr?dinger Equation with A Variable Coefficient,” SIAM J Numer Anal, Vol. 29, 1992, pp. 174-181. doi:10.1137/0729011
[13] W. Dai, G. Li, R. Nassar and S. Su, “On the Stability of the FDTD Method for Solving A Time-Dependent Schr?dinger Equation,” Numer Methods Partial Differential Eq. Vol. 21, 2005, pp. 1140-1154. doi:10.1002/num.20082
[14] S. Descomb and M. Thalhammer, “An Exact Local Error Representation of Exponential Operator Splitting Methods for Evolutionary Problems and Applications to Linear Schr?dinger Equations in the Semi-Classical Regime,” BIT Numer Math, Vol. 50, 2010, pp. 729-749.
[15] T. Fevens and H. Jiang, “Absorbing boundary conditions for the Schr?dinger equation,” SIAM J Sci Comput, Vol. 21, 1999, pp. 255-282. doi:10.1137/S1064827594277053
[16] F. Fujiwara, “FDTD-Q Analysis of a Mott Insulator Quantum Phase,” Ph.D. diss., University of Tokyo, Japan 2007.
[17] H. Han, J. Jin and X. Wu, “A Finite-Difference Method for the One Dimensional Time Dependent Schr?dinger Equation on Unbounded Domain,” Computers and Mathematics with Applications, Vol. 50, 2005, pp. 1345-1362. doi:10.1016/j.camwa.2005.05.006
[18] M. Hochbruuck and C. Lubich, “On Magnus Integrators for Time-Dependent Schr?dinger Equations,” SIAM J Numer Anal, Vol. 41, 2003, pp. 945-963.
[19] B. Jazbi and M. Moini, “On the Numerical Solution of One Dimensional Schr?dinger Equation with Boundary Conditions Involving Fractional Differential Operators,” IUST International Journal of Engineering Science, Vol. 19, 2008, pp. 21-26.
[20] Z. Kalogiratou, Th. Monovasilis and T. E. Simos, “Asymptotically Symplectic Integrators of 3rd and 4th Order for the Numerical Solution of the Schr?dinger Equation,” Computational Fluid and Solid Mechanics 2003, pp. 2012-2015.
[21] K. S. Kunz and R. J. Luebbers, “The Finite Difference Time Domain Method for Electromagnetics,” CRC Press, Roca Raton, FL, 1993.
[22] J. P. Kuska, “Absorbing Boundary Conditions for the Schr?dinger Equation On Finite Intervals,” Physical Review B, Vol. 46, 1992, pp. 5000-5003. doi:10.1103/PhysRevB.46.5000
[23] J. Lo and B. D. Shizgal, “Spectral Convergence of the Quadrature Discretization Method in the Solution of the Schr?dinger and Fokker-Planck Equations: Comparison with Sinc Methods,” J Chem Phys, Vol. 125, 2006, pp. 194108. doi:10.1063/1.2378622
[24] P. L. Nash and L.Y. Chen, “Efficient Finite Difference Solutions to the Time-Dependent Schr?dinger Equation,” J Comput Phys, Vol. 130, 1997, pp. 266-268. doi:10.1006/jcph.1996.5589
[25] J. M. Sanz-Serna, “Methods for the Numerical Solution of the Schr?dinger Equation,” Math Comp, Vol. 43, 1984, pp. 21-27. doi:10.1090/S0025-5718-1984-0744922-X
[26] T. Shibata, “Absorbing Boundary Conditions for Finite-Difference Time-Domain Calculation of the One-Dimensional Schr?dinger Equation,” Physical Review B, Vol. 42, 1991, pp. 6760-6763. doi:10.1103/PhysRevB.43.6760
[27] A. Soriano, E. A. Navarro, J. A. Porti and V. Such, “Analysis of the Finite Difference Time Domain Technique to Solve the Schr?dinger Equation for Quantum Devices,” Journal of Applied Physics, Vol. 95, 2004, pp. 8011-8018. doi:10.1063/1.1753661
[28] M. Stricklans and D. Yager-Elorriaga, “A Parallel Algorithm for Solving the 3D Schr?dinger Equation,” arXiv.0904.0939v4 4 June 2010.
[29] D. M. Sullivan and D. S. Citrin, “Time-domain Simulation of Two Electrons in A Quantum Dot,” Journal of Applied Physics, Vol. 89, 2001, pp. 3841-3846. doi:10.1063/1.1352559
[30] D. M. Sullivan and D. S. Citrin, “Time-domain Simulation of A Universal Quantum Gate,” Journal of Applied Physics Vol. 96, 2002, pp. 3219-3226. doi:10.1063/1.1445277
[31] J. Szeftel, “Design of Absorbing Boundary Condition for Schr?dinger Equations in Rd,” SIAM J Numer Anal Vol. 42, 2004, pp. 1527-1551. doi:10.1137/S0036142902418345
[32] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain,” Artech House, Boston, 1995.
[33] L. Wu, “Dufort-Frankel-Type Methods for Linear and Nonlinear Schr?dinger Equations,” SIAM J Numer Anal, Vol. 33, 1996, pp. 1526-1533. doi:10.1137/S0036142994270636
[34] K. W. Morton and D. F. Mayers, “Numerical Solution of Partial Differential Equations,” Cambridge University Press, London, 1994.

  
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