Year: 2022
Author: Renato Spigler
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1341–1361
Abstract
The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic) Schrödinger equation. We point out that the qlB method actually approximates the hyperbolic version of the non-relativistic Schrödinger equation, whose solution is thus obtained at the price of an additional small error. Such an error is of order of $(ω_c\tau)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $\tau$ a chosen reference time (i.e., 1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundary layer and, consequently, the approximation holds uniformly, which fact is relevant in view of numerical approximations. We also discuss this occurrence more generally, for some classes of linear singularly perturbed partial differential equations.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2022-0001
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1341–1361
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Schrödinger equation hyperbolic Schrödinger equation Dirac equations quantum Lattice Boltzmann Klein-Gordon equation singular perturbations.
Author Details
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High-frequency magnetohydrodynamics
Spigler, Renato
Physics Letters A, Vol. 481 (2023), Iss. P.129013
https://doi.org/10.1016/j.physleta.2023.129013 [Citations: 0]