Algebraic Construction of Spherical Harmonics

The angular wave functions for a hydrogen atom are well known to be spherical harmonics, and are obtained as the solutions of a partial differential equation. However, the differential operator is given by the Casimir operator of the $SU(2)$ algebra and its eigenvalue $l(l+1) \hbar^2$, where $l$ is non-negative integer, is easily obtained by an algebraic method. Therefore the shape of the wave function may also be obtained by extending the algebraic method. In this paper, we describe the method and show that wave functions with different quantum numbers are connected by a rotational group in the cases of $l=0$, 1 and 2.