dorsal/arxiv
View SchemaAccurate numerical solutions of the time-dependent Schr\"odinger equation
| Authors | W. van Dijk, F. M. Toyama |
|---|---|
| Categories | |
| ArXiv ID | physics/0701150 |
| URL | https://arxiv.org/abs/physics/0701150 |
| DOI | 10.1103/PhysRevE.75.036707 |
| Journal | Phys. Rev. E 75, 036707 (2007) |
Abstract
We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schr\"odinger equation. The generalization yields numerical solutions accurate to order $(\Delta x)^{2r-1}$ in space and $(\Delta t)^{2M}$ in time for any positive integers $r$ and $M$, while CN employ $r=M=1$. We note dramatic improvement in the attainable precision (circa 10 or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wavepacket scattering, and long-time studies of decaying systems.
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"abstract": "We present a generalization of the often-used Crank-Nicolson (CN) method of\nobtaining numerical solutions of the time-dependent Schr\\\"odinger equation. The\ngeneralization yields numerical solutions accurate to order $(\\Delta x)^{2r-1}$\nin space and $(\\Delta t)^{2M}$ in time for any positive integers $r$ and $M$,\nwhile CN employ $r=M=1$. We note dramatic improvement in the attainable\nprecision (circa 10 or greater orders of magnitude) along with several orders\nof magnitude reduction of computational time. The improved method is shown to\nlead to feasible studies of coherent-state oscillations with additional\nshort-range interactions, wavepacket scattering, and long-time studies of\ndecaying systems.",
"arxiv_id": "physics/0701150",
"authors": [
"W. van Dijk",
"F. M. Toyama"
],
"categories": [
"physics.comp-ph"
],
"doi": "10.1103/PhysRevE.75.036707",
"journal_ref": "Phys. Rev. E 75, 036707 (2007)",
"title": "Accurate numerical solutions of the time-dependent Schr\\\"odinger equation",
"url": "https://arxiv.org/abs/physics/0701150"
},
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