- 期刊
Comparison of Methods for Solving the Schrödinger Equation for Multiquantum Well Heterostructure Applications
Schrödingeri võrrandi lahendusmeetodite võrdlus mitmik-kvantaukudega heterostruktuuride jaoks
並列摘要
Direct numerical approaches for the solution of the time-independent one-dimensional Schrödinger equation are discussed. Applications to multiquantum well (MQW) semiconductor heterostructure potentials need linear dependence of the computer time t(superscript comp) on the number of spatial grid points N. Although acknowledged as a very effective Fourier grid Hamiltonian (FGH) method, it has cubic dependence on the number of spatial grid points, i.e., t(superscript comp)~N^3, which limits its use to problems with a complexity of N≤1000. A simple straightforward shooting method (ShM), which is based on trial stepping over the coordinate and energy, has the necessary t(superscript comp) ~N dependence with moderate energy convergence efficiency but the recommended symmetry preconditions and the not very clearly defined external boundaries make its application inconvenient. This paper offers a new reliable and effective energy and wave function coupled solution (EWC) method with a Newton iteration scheme and an internal bordered tridiagonal matrix solver. The method has a linear t(superscript comp)~N dependence and may by applied to arbitrary potential energy distribution tasks with complexity up to N=10^5 and beyond. Zero or cyclic boundary conditions may be specified for the wave function. For versatile MQW tasks the combined use of ShM and EWC is illustrated. Detailed accuracy and computer time comparisons show that the combined ShM+EWC method is three orders of magnitude more effective than the FGH method.
延伸閱讀
- Kannike, A. (2006). KULTUURILISE JÄRJEPIDEVUSE HOIDMINE KODUSFÄÄRIS. NÕUKOGUDE EESTI NÄIDE. Acta Historica Tallinnensia, 10(1), 212-229. https://www.airitilibrary.com/Article/Detail?DocID=17367476-200612-201011020001-201011020001-212-229
- Koppel, T., Ainola, L., & Puust, R. (2007). Peatorustiku ja veevõrgu lekete määramise matemaatiline mudel. Proceedings of the Estonian Academy of Sciences.Engineering, 13(1), 3-16. https://www.airitilibrary.com/Article/Detail?DocID=14060175-200732-201011040080-201011040080-3-16
- Kaldmäe, A., Kotta, U., Shumsky, A., & Zhirabok, A. (2016). Häiringu kompenseerimine mõõdetava tagasisidega: sensorite paiknemine. Proceedings of the Estonian Academy of Sciences, 65(4), 317-329. https://doi.org/10.3176/proc.2016.4.05
- Pall, P., Vilbaste, S., Kõiv, T., Kõrs, A., Käiro, K., Laas, A., Nõges, P., Nõges, T., Piirsoo, K., Toomsalu, L., & Viik, M. (2011). Süsiniku ja toitesoolade koormus Võrtsjärve sissevooludes ning väljavoolus. Estonian Journal of Ecology, 60(1), 39-53. https://www.airitilibrary.com/Article/Detail?DocID=17367549-201103-201104080006-201104080006-39-53
- Barashkova, T. (2006). Korrelatsiooni uurimine vahelduvpinge mõõtmisel. Proceedings of the Estonian Academy of Sciences.Engineering, 12(3-1), 208-217. https://www.airitilibrary.com/Article/Detail?DocID=14060175-200609-201011040077-201011040077-208-217