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 $\hat L_z$).
Highlights
Stasyuk’s career has been spent working with operator formalisms and finding different ways to perform calculations or simplify complex expressions. It is within this spirit that we present this work, which provides a clear operator-based methodology for determining spherical harmonics
The final approach is to work with simple harmonic oscillator wavefunctions and the angular momentum representation discovered by Schwinger [19]
We show how one can derive spherical harmonics without using any derivatives and argue that this approach brings in a number of important pedagogical advantages to the conventional treatment of orbital angular momentum
Summary
Stasyuk’s career has been spent working with operator formalisms (especially Hubbard X-operators) and finding different ways to perform calculations or simplify complex expressions. It is within this spirit that we present this work, which provides a clear operator-based methodology for determining spherical harmonics. The motivation for this work is a book project of one of the authors (JKF) on teaching quantum mechanics without calculus [1]. This project is a follow-up of a massively open on-line course called Quantum Mechanics for Everyone, which is running on edX from April 2017–April 2019 [2]. We end with a discussion of how this approach can be extended to determine the rotation matrices and provide some commentary about pedagogy
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