Calculating spherical harmonics without derivatives

TL;DR

This paper introduces a new operator-based method utilizing the exponential disentangling operator identity to derive spherical harmonics more naturally, simplifying the process for students and educators in quantum mechanics.

Contribution

It presents a novel approach to deriving spherical harmonics using operators and the exponential disentangling operator identity, making the derivation more accessible.

Findings

Simplifies the derivation of spherical harmonics

Provides a more natural and intuitive method

Connects quantum optics techniques to quantum mechanics education

Abstract

The derivation of spherical harmonics is the same in nearly every quantum mechanics textbook and classroom. It is found to be difficult to follow, hard to understand, and challenging to reproduce by most students. In this work, we show how one can determine spherical harmonics in a more natural way based on operators and a powerful identity called the exponential disentangling operator identity (known in quantum optics, but little used elsewhere). This new strategy follows naturally after one has introduced Dirac notation, computed the angular momentum algebra, and determined the action of the angular momentum raising and lowering operators on the simultaneous angular momentum eigenstates (under $\hat L^2$ and $\hat L_z$).