Obtenção dos Harmônicos Hiperesféricos em N Dimensões
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Tipo de documento: | Trabalho de conclusão de curso |
Idioma: | por |
Título da fonte: | Repositório Institucional da UFS |
Texto Completo: | http://ri.ufs.br/jspui/handle/riufs/12842 |
Resumo: | In this work we build hyperspherical harmonics in N dimensions. The Laplace equation in N dimensions in hyperspherical coordinates is obtained using the Laplace-Beltrami operator with the spherical geometry metric. The method used to obtain hyperspherical harmonics is based on the usual method of separating variables and does not involve harmonic polynomial theory, generalized angular momentum theory, or group representation theory. Ordinary equations are reduced to the Schr¨odinger equation with the P¨oschl-Teller symmetrical potential. The solutions of the ordinary equations are presented in the form of the solution of the Schr¨odinger equation multiplied by a functional factor computed in the solution process. Hyperspherical harmonics are obtained as the product of solutions of ordinary equations and are expressed in terms of Gegenbauer polynomials. The result is compared with the results obtained by other methods. For the graphical illustration of the results are presented the images of the projections of the 4-dimensional hyperspherical harmonics in the three-dimensional hyperplanes. |
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Santos, Aleff de JesusSmirnov, Andrei2020-02-28T19:31:28Z2020-02-28T19:31:28Z2019-12-20SANTOS, Aleff de Jesus. Obtenção dos Harmônicos Hiperesféricos em N Dimensões. São Cristóvão, SE, 2019. Monografia – Departamento de Física, Centro de Ciências Exatas e Tecnologia, Universidade Federal de Sergipe, São Cristóvão, 2019.http://ri.ufs.br/jspui/handle/riufs/12842In this work we build hyperspherical harmonics in N dimensions. The Laplace equation in N dimensions in hyperspherical coordinates is obtained using the Laplace-Beltrami operator with the spherical geometry metric. The method used to obtain hyperspherical harmonics is based on the usual method of separating variables and does not involve harmonic polynomial theory, generalized angular momentum theory, or group representation theory. Ordinary equations are reduced to the Schr¨odinger equation with the P¨oschl-Teller symmetrical potential. The solutions of the ordinary equations are presented in the form of the solution of the Schr¨odinger equation multiplied by a functional factor computed in the solution process. Hyperspherical harmonics are obtained as the product of solutions of ordinary equations and are expressed in terms of Gegenbauer polynomials. The result is compared with the results obtained by other methods. For the graphical illustration of the results are presented the images of the projections of the 4-dimensional hyperspherical harmonics in the three-dimensional hyperplanes.Constru´ımos neste trabalho os harmˆonicos hiperesf´ericos em N dimens˜oes. A equa- ¸c˜ao de Laplace em N dimens˜oes nas coordenadas hiperesf´ericas ´e obtida com o uso do operador de Laplace-Beltrami com a m´etrica da geometria esf´erica. O m´etodo usado para a obten¸c˜ao dos harmˆonicos hiperesf´ericos ´e baseado no m´etodo usual de separa¸c˜ao de vari´aveis e n˜ao envolve a teoria de polinˆomios harmˆonicos, teoria de momento angular generalizado ou teoria de representa¸c˜ao de grupo. As equa¸c˜oes ordin´arias s˜ao reduzidas equa¸c˜ao de Schr¨odinger com o potencial sim´etrico de P¨oschl-Teller. As solu¸c˜oes das equa¸c˜oes ordin´arias s˜ao apresentadas na forma da solu¸c˜ao da equa¸c˜ao de Schr¨odinger multiplicada por um fator funcional computado no processo de solu¸c˜ao. Os harmˆonicos hiperesf´ericos s˜ao obtidos como o produto das solu¸c˜oes das equa¸c˜oes ordin´arias e s˜ao expressos em termos dos polinˆomios de Gegenbauer. O resultado ´e comparado com os resultados obtidos por outros m´etodos. Para a ilustra¸c˜ao gr´afica dos resultados s˜ao apresentadas as imagens das proje¸c˜oes dos harmˆonicos hiperesf´ericos em 4 dimens˜oes nos hiperplanos tridimensionais.São Cristóvão, SEporFísicaEnsino de FísicaOperador de Laplace-BeltramiHarmônicos HiperesféricosPolinômios de GegenbauerLaplace-Beltrami OperatorHyperspherical HarmonicsGegenbauer PolynomialsCIENCIAS EXATAS E DA TERRA::FISICA::FISICA GERALObtenção dos Harmônicos Hiperesféricos em N Dimensõesinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/bachelorThesisUniversidade Federal de SergipeDFI - Departamento de Física – São Cristóvão - Presencialreponame:Repositório Institucional da UFSinstname:Universidade Federal de Sergipe (UFS)instacron:UFSinfo:eu-repo/semantics/openAccessLICENSElicense.txtlicense.txttext/plain; charset=utf-81475https://ri.ufs.br/jspui/bitstream/riufs/12842/1/license.txt098cbbf65c2c15e1fb2e49c5d306a44cMD51ORIGINALAleff_Jesus_Santos.pdfAleff_Jesus_Santos.pdfapplication/pdf4925246https://ri.ufs.br/jspui/bitstream/riufs/12842/2/Aleff_Jesus_Santos.pdf61e7b3ef3e1621cb9d484bd64456cc33MD52TEXTAleff_Jesus_Santos.pdf.txtAleff_Jesus_Santos.pdf.txtExtracted texttext/plain104747https://ri.ufs.br/jspui/bitstream/riufs/12842/3/Aleff_Jesus_Santos.pdf.txt3a6d2504e9940e14665a3d016998d114MD53THUMBNAILAleff_Jesus_Santos.pdf.jpgAleff_Jesus_Santos.pdf.jpgGenerated Thumbnailimage/jpeg1279https://ri.ufs.br/jspui/bitstream/riufs/12842/4/Aleff_Jesus_Santos.pdf.jpg596c5feb4af12e7b426b7d314928b926MD54riufs/128422020-02-28 16:31:28.868oai:ufs.br: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Repositório InstitucionalPUBhttps://ri.ufs.br/oai/requestrepositorio@academico.ufs.bropendoar:2020-02-28T19:31:28Repositório Institucional da UFS - Universidade Federal de Sergipe (UFS)false |
dc.title.pt_BR.fl_str_mv |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
title |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
spellingShingle |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões Santos, Aleff de Jesus Física Ensino de Física Operador de Laplace-Beltrami Harmônicos Hiperesféricos Polinômios de Gegenbauer Laplace-Beltrami Operator Hyperspherical Harmonics Gegenbauer Polynomials CIENCIAS EXATAS E DA TERRA::FISICA::FISICA GERAL |
title_short |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
title_full |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
title_fullStr |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
title_full_unstemmed |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
title_sort |
Obtenção dos Harmônicos Hiperesféricos em N Dimensões |
author |
Santos, Aleff de Jesus |
author_facet |
Santos, Aleff de Jesus |
author_role |
author |
dc.contributor.author.fl_str_mv |
Santos, Aleff de Jesus |
dc.contributor.advisor1.fl_str_mv |
Smirnov, Andrei |
contributor_str_mv |
Smirnov, Andrei |
dc.subject.por.fl_str_mv |
Física Ensino de Física Operador de Laplace-Beltrami Harmônicos Hiperesféricos Polinômios de Gegenbauer |
topic |
Física Ensino de Física Operador de Laplace-Beltrami Harmônicos Hiperesféricos Polinômios de Gegenbauer Laplace-Beltrami Operator Hyperspherical Harmonics Gegenbauer Polynomials CIENCIAS EXATAS E DA TERRA::FISICA::FISICA GERAL |
dc.subject.eng.fl_str_mv |
Laplace-Beltrami Operator Hyperspherical Harmonics Gegenbauer Polynomials |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::FISICA::FISICA GERAL |
description |
In this work we build hyperspherical harmonics in N dimensions. The Laplace equation in N dimensions in hyperspherical coordinates is obtained using the Laplace-Beltrami operator with the spherical geometry metric. The method used to obtain hyperspherical harmonics is based on the usual method of separating variables and does not involve harmonic polynomial theory, generalized angular momentum theory, or group representation theory. Ordinary equations are reduced to the Schr¨odinger equation with the P¨oschl-Teller symmetrical potential. The solutions of the ordinary equations are presented in the form of the solution of the Schr¨odinger equation multiplied by a functional factor computed in the solution process. Hyperspherical harmonics are obtained as the product of solutions of ordinary equations and are expressed in terms of Gegenbauer polynomials. The result is compared with the results obtained by other methods. For the graphical illustration of the results are presented the images of the projections of the 4-dimensional hyperspherical harmonics in the three-dimensional hyperplanes. |
publishDate |
2019 |
dc.date.issued.fl_str_mv |
2019-12-20 |
dc.date.accessioned.fl_str_mv |
2020-02-28T19:31:28Z |
dc.date.available.fl_str_mv |
2020-02-28T19:31:28Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/bachelorThesis |
format |
bachelorThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
SANTOS, Aleff de Jesus. Obtenção dos Harmônicos Hiperesféricos em N Dimensões. São Cristóvão, SE, 2019. Monografia – Departamento de Física, Centro de Ciências Exatas e Tecnologia, Universidade Federal de Sergipe, São Cristóvão, 2019. |
dc.identifier.uri.fl_str_mv |
http://ri.ufs.br/jspui/handle/riufs/12842 |
identifier_str_mv |
SANTOS, Aleff de Jesus. Obtenção dos Harmônicos Hiperesféricos em N Dimensões. São Cristóvão, SE, 2019. Monografia – Departamento de Física, Centro de Ciências Exatas e Tecnologia, Universidade Federal de Sergipe, São Cristóvão, 2019. |
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Universidade Federal de Sergipe |
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DFI - Departamento de Física – São Cristóvão - Presencial |
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