Accurate Quantum States for a 2D-Dipole
Edge dislocations are crucial in understanding both mechanical and electrical transport in solid and are modeled as line distributions of dipole moments. The calculation of the electronic spectrum for the two dimensional dipole, represented by the potential energy <inline-formula><math xmln...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Nanomaterials |
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| Online Access: | https://www.mdpi.com/2079-4991/14/2/206 |
| _version_ | 1797342871964090368 |
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| author | Daniel Vrinceanu |
| author_facet | Daniel Vrinceanu |
| author_sort | Daniel Vrinceanu |
| collection | DOAJ |
| description | Edge dislocations are crucial in understanding both mechanical and electrical transport in solid and are modeled as line distributions of dipole moments. The calculation of the electronic spectrum for the two dimensional dipole, represented by the potential energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo form="prefix">cos</mo><mi>θ</mi><mo>/</mo><mi>r</mi></mrow></semantics></math></inline-formula>, has been the topic of several studies that show significant difficulties in obtaining accurate results. In this work, we demonstrate that the source of these difficulties is a logarithmic contribution to the behavior of the wave function at the origin that was neglected by previous authors. By taking into account this non-analytic deviation of the solution of Schrödinger’s equation, superior results, with the expected rate of convergence, are obtained. This goal is accomplished by “adapting” general algorithms for solving partial derivative differential equations to include the desired asymptotic behavior. We illustrate this principle for the variational principle and finite difference methods. Accurate energies and wave functions are obtained not only for the ground state but also for the first eleven excited states and are useful for designing nanoelectronic devices. This paper demonstrates that augmentary knowledge about analytic properties of the solutions leads to the improved convergence and stability of numerical methods. |
| first_indexed | 2024-03-08T10:39:27Z |
| format | Article |
| id | doaj.art-86ddecb2f4774c5b8849d2e78d4cd833 |
| institution | Directory Open Access Journal |
| issn | 2079-4991 |
| language | English |
| last_indexed | 2024-03-08T10:39:27Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Nanomaterials |
| spelling | doaj.art-86ddecb2f4774c5b8849d2e78d4cd8332024-01-26T17:58:39ZengMDPI AGNanomaterials2079-49912024-01-0114220610.3390/nano14020206Accurate Quantum States for a 2D-DipoleDaniel Vrinceanu0Department of Physics, Texas Southern University, Houston, TX 77004, USAEdge dislocations are crucial in understanding both mechanical and electrical transport in solid and are modeled as line distributions of dipole moments. The calculation of the electronic spectrum for the two dimensional dipole, represented by the potential energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo form="prefix">cos</mo><mi>θ</mi><mo>/</mo><mi>r</mi></mrow></semantics></math></inline-formula>, has been the topic of several studies that show significant difficulties in obtaining accurate results. In this work, we demonstrate that the source of these difficulties is a logarithmic contribution to the behavior of the wave function at the origin that was neglected by previous authors. By taking into account this non-analytic deviation of the solution of Schrödinger’s equation, superior results, with the expected rate of convergence, are obtained. This goal is accomplished by “adapting” general algorithms for solving partial derivative differential equations to include the desired asymptotic behavior. We illustrate this principle for the variational principle and finite difference methods. Accurate energies and wave functions are obtained not only for the ground state but also for the first eleven excited states and are useful for designing nanoelectronic devices. This paper demonstrates that augmentary knowledge about analytic properties of the solutions leads to the improved convergence and stability of numerical methods.https://www.mdpi.com/2079-4991/14/2/206edge dislocationselectronic statesnumerical methods |
| spellingShingle | Daniel Vrinceanu Accurate Quantum States for a 2D-Dipole Nanomaterials edge dislocations electronic states numerical methods |
| title | Accurate Quantum States for a 2D-Dipole |
| title_full | Accurate Quantum States for a 2D-Dipole |
| title_fullStr | Accurate Quantum States for a 2D-Dipole |
| title_full_unstemmed | Accurate Quantum States for a 2D-Dipole |
| title_short | Accurate Quantum States for a 2D-Dipole |
| title_sort | accurate quantum states for a 2d dipole |
| topic | edge dislocations electronic states numerical methods |
| url | https://www.mdpi.com/2079-4991/14/2/206 |
| work_keys_str_mv | AT danielvrinceanu accuratequantumstatesfora2ddipole |