Time-dependent Schrödinger Equation based on HO-FDTD Schemes
Keywords:
HO-FDTD, numerical dispersion, stability, the Schrödinger equationAbstract
A high order finite-different time-domain methods using Taylor series expansion for solving time-dependent Schrödinger equation has been systematically discussed in this paper. Numerical characteristics have been investigated of the schemes for the Schrödinger equation. Compared with the standard Yee FDTD scheme, the numerical dispersion has been decreased and the convergence has been improved. The general update equations of the methods have been presented for wave function. Numerical results of potential well in one-dimension show that the application of the schemes is more effective than the Yee’s FDTD method and the higher order has the better numerical dispersion characteristics.
Downloads
References
S. Datta, Quantum Transport: Atom to Transistor, Cambridge University Press, New York, 2005.
A. Z. Elsherbeni, FDTD Course Notes, Department of Electrical Engineering, The University of Mississippi, MS; Spring 2001.
K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equation in isotropic media,” IEEE Trans. Antennas Propag. vol. AP-14, pp. 302-307, 1966.
D. M. Sullivan, Electromagnetic Simulation using the FDTD Method, IEEE Press, New York, 2000.
C. W. Manry, S. L. Broschat, and J. B. Schneider, “High-order FDTD methods for large problems,” Applied Computational Electromagnetics Society, vol. 10, no. 2, pp.17-29, 1995.
A. Goldberg, H. M. Schey, and J. L. Schwartz “Computer-generated motion pictures of one dimensional quantum mechanical transmission and reflection phenomena,” American Journal of Physics, vol. 35, no. 3, pp.177-186, 1967.
T. Shibata, “Absorbing boundary conditions for the finite-difference time-domain calculation of the one dimensional Schrödinger equation,” Physical Review B, vol. 42, no.8, pp. 6760-6763, 1991.
J. P. Kuska, “Absorbing boundary conditions for the Schrödinger equation on finite intervals,” Physical Review B, vol. 46, no. 8, pp. 5000-5003, 1992.
C. J. Ryu, A. Y. Liu, W. E. I. Sha, and C. W. Chew, “Finite-difference time-domain simulation of the Maxwell-Schrodinger system,” IEEE Journal on Multiscale and Multiphysics Comput. Techniques, vol. 1, pp. 1-8, 2016.
J. Houle, D. Sullivan, E. Crowell, S. Mossman, and M. G. Kuzyk, “Three dimensional time domain simulation of the quantum magnetic susceptibility,” IEEE Workshop on Microelectronics and Electron Devices, 2019.
K. Lan, Y. W. Li, and W. G. Lin, “A High order (2, 4) scheme for reducing dispersion in FDTD algorithm,” IEEE Trans. Electromagn. Compat., vol. 41, no. 2, pp. 160-165, 1995.
M. Krumpholz and L. P. B. Katehi, “MRTD: New time-domain schemes based on multiresolution analysis,” IEEE Trans. on Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-571, 1996.
G. Xie, Z. X. Huang, and W. E. I. Sha, “Simulating Maxwell-Schrödinger equations by high-order symplectic FDTD algorithm,” IEEE Journal on Multiscal and Multiphysics Comput. Tech., vol. 4, pp. 143-151, 2019.
Q. S. Cao, R. Kanapady, and F. Reitich, “Highorder Runge-Kutta multiresolution time-domain methods for computational electromagnetics,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3316-3326, 2006.
G. Sun and C. W. Trueman, “Analysis and numerical experiments on the numerical dispersion of two-dimensional,” IEEE Antenna and Wireless Propag. Lett., vol. 2, no. 7, pp. 78-81, 2003.
M. Zhu and Q. S. Cao, “Analysis for threedimensional curved objects by Runge-kutta high order time-domain method,” Applied Computational Electromagnetics Society, vol. 30, no. 1, pp. 86-92, 2015.
M. Zhu and Q. S. Cao, “Studying and analysis of the characteristic of the high-order and MRTD and RK-MRTD scheme,” Applied Computational Electromagnetics Society, vol. 28, no. 5, pp. 380- 389, 2013.
S. Gottlieb, C. W. Shu, and E. Tadmor, “Strong stability Strong stability-preserving high-order time discretization methods,” SIAM Rev., vol. 43, no. 1, pp. 89-112, 2001.
J. Shen, W. E. I Sha, Z. X. Huang, M. S. Chen, and X. L. Wu, “High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation,” Comput. Physics Comm., vol. 184, no. 3, pp. 480- 492, 2013.
