Quantum Mechanics: Concepts and Applications · Problem 5.26

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · ladder-operator-commutators-spherical

Problem

Using the expression of L^−\hat{L}_- in spherical coordinates, prove the following two commutators: [L^−, e−iφsin⁡θ]=0and[L^−, cos⁡θ]=ℏe−iφsin⁡θ.[\hat{L}_-,\ e^{-i\varphi}\sin\theta] = 0 \quad \text{and} \quad [\hat{L}_-,\ \cos\theta] = \hbar e^{-i\varphi}\sin\theta.

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