Obtenção dos Harmônicos Hiperesféricos em N Dimensões

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.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, S