Abstract
Numerical solutions for laser–matter interaction in Schrödinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small cost, to extend any fourth-order scheme for Schrödinger equation with time-independent potential to a fourth-order method for Schrödinger equation with laser potential. This is made possible due to the highly specific form of the time dependent potential which is linear in space in the case of laser–matter interaction. This leads to a highly amenable structure of the commutators in the Magnus expansion, where, in particular, the first commutator reduces to a scalar multiple of the gradient. In turn, this special structure allows us to split the Magnus expansion effectively via a variety of fourth-order splittings. Additionally, the error constant remains tiny because we keep the integrals of the potential intact to the very last stage of computations. This is particularly important in the case of highly oscillatory potentials. As demonstrated via numerical examples, these fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.
Original language | English |
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Pages (from-to) | 195-201 |
Number of pages | 7 |
Journal | Computer Physics Communications |
Volume | 234 |
Early online date | 20 Jul 2018 |
DOIs | |
Publication status | Published - 1 Jan 2019 |
Funding
The work of Karolina Kropielnicka in this project was financed by The National Center for Science , based on Grant No. 2016/22/M/ST1/00257 .
Keywords
- Compact methods
- Higher order method
- Laser potential
- Magnus expansion
- Schrödinger equation
- Splitting methods
- Time dependent potentials
ASJC Scopus subject areas
- Hardware and Architecture
- General Physics and Astronomy
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Pranav Singh
- Department of Mathematical Sciences - Senior Lecturer
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
- Institute for Mathematical Innovation (IMI)
Person: Research & Teaching, Core staff