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2018-03-27
Symplectic Pseudospectral Time-Domain Scheme for Solving Time-Dependent Schrodinger Equation
By
Progress In Electromagnetics Research M, Vol. 66, 109-118, 2018
Abstract
A symplectic pseudospectral time-domain (SPSTD) scheme is developed to solve Schrodinger equation. Instead of spatial finite differences in conventional finite-difference time-domain (FDTD) methods, fast Fourier transform is used to calculate spatial derivatives. In time domain, the scheme adopts high-order symplectic integrators to simulate time evolution of Schrodinger equation. A detailed numerical study on the eigenvalue problems of 1D quantum well and 3D harmonic oscillator is carried out. The simulation results strongly confirm the advantages of the SPSTD scheme over the traditional PSTD method and FDTD approach. Furthermore, by comparing to the traditional PSTD method and the non-symplectic Runge-Kutta (RK) method, the explicit SPSTD scheme, which is an infinite order of accuracy in space domain and energy-conserving in time domain, is well suited for a long-term simulation.
Citation
Jing Shen, Wei E. I. Sha, Xiaojing Kuang, Jinhua Hu, Zhixiang Huang, and Xian-Liang Wu, "Symplectic Pseudospectral Time-Domain Scheme for Solving Time-Dependent Schrodinger Equation," Progress In Electromagnetics Research M, Vol. 66, 109-118, 2018.
doi:10.2528/PIERM18010808
References

1. Datta, S., Quantum Transport: Atom to Transistor, Cambridge University Press, 2005.
doi:10.1017/CBO9781139164313

2. Soriano, A., A. E. Navarro, A. J. Porti, and V. Such, "Analysis of the finite difference time domain technique to solve the Schrodinger equation for quantum devices," J. Appl. Phys., Vol. 95, No. 12, 8011-8018, 2004.
doi:10.1063/1.1753661

3. Sullivan, D. M. and D. S. Citrin, "Determining quantum eigenfunctions in three dimensional nanoscale structures," J. Appl. Phys., Vol. 97, No. 10, 581-592, 2005.
doi:10.1063/1.1896437

4. Cai, J. X. and Y. S. Wang, "A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrödinger system," Chin. Phys. B, Vol. 22, No. 6, 135-140, 2013.
doi:10.1088/1674-1056/22/6/060207

5. He, J. P., L. F. Shen, Q. Zhang, and S. L. He, "A pseudospectral time-domain algorithm for calculating the band structure of a two-dimensional photonic crystal," Chin. Phys. Lett., Vol. 19, No. 4, 507-510, 2002.
doi:10.1088/0256-307X/19/4/318

6. Liu, Q. H., "The PSTD algorithm: A time-domain method requiring only two cells per wavelength," Microw. Opt. Technol. Lett., Vol. 15, 158-165, 1997.
doi:10.1002/(SICI)1098-2760(19970620)15:3<158::AID-MOP11>3.0.CO;2-3

7. Brendan, B. G., "Improved numerical cherenkov instability suppression in the generalized PSTD PIC algorithm," Computer Physics Communications, Vol. 196, 221-225, 2015.

8. Mechthild, T. and J. Siam, "Convergenceanalysis of high-order time-spliting pseudospectral methods for nonlinear Schrödinger equation," SIAM J. Numer. Anal., Vol. 50, No. 6, 3231-3258, 2012.
doi:10.1137/120866373

9. Shi, Y. and C. H. Liang, "Analysis of the left-handed metamaterials using multi-domain pseudospectral time-domain algorithm," Progress In Electromagnetics Research, Vol. 51, 153-165, 2005.
doi:10.2528/PIER04092301

10. Sanz, J. M. and M. P. Calvo, Numerical Hamiltonian Problems, Vol. 64, No. 5, 21-24, Mathematics of Computation, 1994.
doi:10.1007/978-1-4899-3093-4

11. Sheu, T. W. H., R. Y. Chung, and J. H. Li, "Development of a symplectic scheme with optimized numerical dispersion-relation equation to solve Maxwell’s equations in dispersive media," Progress In Electromagnetics Research, Vol. 132, 517-549, 2012.
doi:10.2528/PIER12080901

12. Guyenne, P., D. Nicholls, and C. Sulem, Hamiltonian Partial Differential Equations and Applications, Springer, 2015.
doi:10.1007/978-1-4939-2950-4

13. Tao, M., "Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance," Physical Review E, Vol. 94, No. 4, 3303, 2016.
doi:10.1103/PhysRevE.94.043303

14. Sun, Y. and P. S. P. Tse, "Symplectic and multisymplectic numerical methods for Maxwells equations," J. Comput. Phys., Vol. 230, No. 5, 2076-2094, 2011.
doi:10.1016/j.jcp.2010.12.006

15. Monovasilis, T., Z. Kalogiratou, and T. E. Simos, "Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems," Comput. Phys. Commun., Vol. 177, No. 10, 757-763, 2007.
doi:10.1016/j.cpc.2007.05.020

16. Chen, Z. X., X. You, and W. Shi, "Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems," Comput. Phys. Commun., Vol. 183, No. 1, 86-98, 2012.
doi:10.1016/j.cpc.2011.09.002

17. Shen, J., W. Sha, Z. X. Huang, M. S. Chen, and X. L. Wu, "High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation," Comput. Phys. Commun., Vol. 184, 480-492, 2013.
doi:10.1016/j.cpc.2012.09.032

18. Gray, S. K. and D. E. Manolopoulos, "Symplectic integrators tailored to the time-dependent Schrödinger equation," J. Chem. Phys., Vol. 104, No. 18, 7099-7112, 1996.
doi:10.1063/1.471428

19. Yoshida, H., "Construction of higher-order symplectic integrators," Phys. Lett. A, Vol. 150, No. 5, 262-268, 2008.

20. Sha, W., Z. X. Huang, M. S. Chen, and X. L. Wu, "Survey on symplectic finite-difference time-domain schemes for Maxwell’s equations," IEEE Trans. Antennas Propag., Vol. 56, No. 2, 493-500, 2008.
doi:10.1109/TAP.2007.915444