*Forma,* Vol. 32 (Special Issue II), pp. SII17–SII25, 2017

doi:10.5047/forma.2017.sii004

*Original Paper*

## Algebraic Construction of Spherical Harmonics

Naohisa Ogawa

Hokkaido University of Sciences, Sapporo 006-8585, Japan

E-mail address: ogawanao@hus.ac.jp

(Received April 25, 2017; Accepted July 25, 2017)

**Abstract. **
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)*ħ*^{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.

**Keywords: **
Mirror Operator, Spherical Harmonics

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