Obtenção dos Harmônicos Hiperesféricos em N Dimensões

Abstract

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.