Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics

TL;DR

This paper investigates vector operators that modify the angular momentum quantum number l in spherical harmonics, providing methods to generate the entire set of harmonics from minimal states, advancing quantum angular momentum theory.

Contribution

It extends previous work by analyzing operators that shift l in spherical harmonics and demonstrates how to generate all harmonics from minimal states.

Findings

Identified specific states |lm> with minimal l for given m

Established methods to generate the full set of spherical harmonics

Clarified the role of these operators in angular momentum theory

Abstract

Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics |lm>, primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m, specific states |lm> in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is |0,0>. How to use these states to generate whole set of spherical harmonics is illustrated.