Principles of Quantum Mechanics · Problem 12.5.8

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · angular-momentum-coordinate-basis

Problem

*(Optional).* Verify that

L_x \underset{\text{coordinate basis}}{\longrightarrow} i\hbar\left(\sin\phi\,\frac{\partial}{\partial\theta} + \cos\phi\cot\theta\,\frac{\partial}{\partial\phi}\right)

L_y \underset{\text{coordinate basis}}{\longrightarrow} i\hbar\left(-\cos\phi\,\frac{\partial}{\partial\theta} + \sin\phi\cot\theta\,\frac{\partial}{\partial\phi}\right)