https://doi.org/10.1140/epjb/s10051-024-00759-4
Regular Article - Solid State and Materials
Solving Schrödinger equation within arbitrary spherical quantum dots with neural network
1
Preparatory Institute for Engineering Studies of Kairouan (I.P.E.I.K.), University of Kairouan, Kairouan, Tunisia
2
Laboratory of Chemistry, Materials and Modelling (LR24ES02), Department of Physics, University of Kairouan, Kairouan, Tunisia
3
Research Institute for Applied Physics and Astronomy, University of Tabriz, 51665-163, Tabriz, Iran
Received:
19
June
2024
Accepted:
24
July
2024
Published online:
3
August
2024
In this study, we delve into the realm of solving the Schrödinger equation within spherical quantum dots (QDs) characterized by arbitrary potentials, leveraging the capabilities of machine learning methodologies. Our approach involves training neural networks (NNs) through a curated collection of potentials and wave functions (WFs), which were initially computed using the classical finite element method. To gauge the reliability of the estimates produced by these NNs, we introduce accuracy indicators for rigorous assessment. The training procedure relies on the gradient descent method to optimize the networks’ performance. Furthermore, our investigation encompasses scenarios with analytical potentials, broadening the scope of our analysis beyond empirical cases. By integrating analytical potentials into our study, we aim to achieve a comprehensive understanding of the neural network’s effectiveness in handling various potential profiles. This expansion opens avenues for more versatile and insightful quantum mechanical explorations within the realm of nanoscale systems. Among the findings, the QD core exhibited the highest level of accuracy in WF estimation, achieved through the utilization of a spherical potential. Conversely, the estimation performance was least reliable in scenarios involving HLP, with a notable deviation of 16.68%. Transitioning to the core/shell structure, employing the double HLP configuration resulted in the most precise estimation of WFs. This contrasts significantly with the estimation performance for the V-Shaped Potential, where accuracy was comparatively lower with deviation of 4%.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
