A Modern Approach to Quantum Mechanics · Problem 9.6

Translational and Rotational Symmetry in the Two-Body Problem — John S. Townsend

Physics — Quantum Mechanics · vector-operator-rotation-commutation

Problem

Show that the commutation relations of the orbital angular momentum operators can be cast in a similar form as for

\hat{\mathbf{r}}

and

\hat{\mathbf{p}}

under rotations:

[\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y \qquad [\hat{L}_z, \hat{L}_y] = -i\hbar \hat{L}_x \qquad [\hat{L}_z, \hat{L}_z] = 0

All vector operators must behave the same way when they are rotated.