Angular Momentum is conserved iff the potential is invariant under rotations

This should follow from the Lie algebra / Lie group properties of quantum systems, since Angular Momentum is the generator of rotations. I need to clarify this statement.

Proof

• Write out this proof

Sketch:

1. Map the impact of Potential invariance to the Hamiltonian
2. Use the Liouville–von Neumann equation to calculate $\frac{d\rho }{dt}$.
3. Use the above to show that angular momentum $L_z\rho$ is constant, i.e. $\frac{d}{dt}(L_z\rho )=0$