Abstract
We present a new discretisation scheme for the Schrödinger equation based on analytic solutions to local linearisations. The scheme generates the normalised eigenfunctions and eigenvalues simultaneously and is exact for piecewise constant potentials and effective masses. Highly accurate results can be obtained with a small number of mesh points and a robust and flexible algorithm using continuation techniques is derived. An application to the Hartree approximation for SiGe heterojunctions is discussed in which we solve the coupled Schrödinger-Poisson model problem selfconsistently.
Similar content being viewed by others
References
L. Gr. Ixaru and M. Rizea, J. Comp. Phys.,73, 306 (1987).
P. B. Bailey, J. SIAM Appl. Math.,14, 242 (1966).
T. Ando and S. Mori, J. Phys. Soc. Japan,47, 1518 (1979).
T. Ando, J. Phys. Soc. Japan,51, 3900 (1982).
I. H. Tan, G. L. Snider, L. D. Chang and E. L. Hu, J. Appl. Phys.,68, 4071 (1990).
K. Inoue, H. Sakaki, J. Yoshino and T. Hotta, J. Appl. Phys.,58, 4277 (1985).
T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys.,54, 437 (1982).
G. Bastard,Wave Mechanics Applied to Semiconductor Heterostructures, Les éditions de physique 1988.
M. F. H. Schuurmans and G. W.'t Hooft, Phys. Rev.,B31, 8041 (1985).
D. J. Ben Daniel and C. B. Duke, Phys. Rev.,98, 368 (1955).
T. Ando, J. Phys. Soc. Japan,51, 3893 (1982).
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Dover 1972.
S. M. Sze,Physics of Semiconductor Devices, Wiley 1981.
R. People, IEEE,QE-22, 1696 (1986).
Author information
Authors and Affiliations
Additional information
To the memory of my good friend Erik van Loon, taken from us so soon
Rights and permissions
About this article
Cite this article
Geurts, B.J. A new finite difference scheme adapted to the one-dimensional Schrödinger equation. Z. angew. Math. Phys. 44, 654–672 (1993). https://doi.org/10.1007/BF00948481
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00948481