Solution of spherical equation in 3 dimensions for hydrogen atom with quantum numbers 4≤ n≤ 5

Abstract

The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger’s equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions \({B}_{lm}={({-}1)}^{{(m+|m|)}/2}\sqrt{\frac{2l+1(l-|m|)!}{2\,\,\,(l+|m|)!}}\sqrt{\frac{1}{2\pi }}\) . Based on the normalization conditions for n = 4 a wave function is produced \({Y}_{3,3}(\theta \varphi )={({-}1)}^{(3+|3|/2)}\sqrt{\frac{2.3+1(3-|3|)!}{4\pi\,\,\,(3+|3|)!}}{P}_{3}^{3}(cos\theta ){e}^{3i\varphi }\) , while for n = 5 a wave function is \({Y}_{4,4}(\theta \varphi )={({-}1)}^{(4+|4|/2)}\sqrt{\frac{2.4+1(4-|4|)!}{4\pi\,\,\,(4+|4|)!}}{P}_{4}^{4}(cos\theta ){e}^{4i\varphi }\) .