Algebraic construction of spherical harmonics

TL;DR

This paper presents an algebraic method to construct spherical harmonics for hydrogen atom wave functions, connecting different quantum states through rotational symmetry, offering an alternative to differential equation solutions.

Contribution

It introduces an algebraic approach to derive spherical harmonics and demonstrates their connection via the rotational group for specific quantum numbers.

Findings

Algebraic construction of spherical harmonics is feasible.

Wave functions with different quantum numbers are connected by rotational symmetry.

The method applies explicitly for quantum numbers l=0, 1, and 2.

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 $l(l+1) \hbar^2$, 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.