Rotational invariance

Rotational invariance refers to the fact that after a rotation the new system still follows Schrodinger's equation. That is [R,E-H]=0 for any rotation R. since the rotation does not depend explicitly on time, it commutes with the energy operator. Thus for rotational invariance we must have [R,H]=0.

Since [R,E-H]=0, and because for infinitesimal rotation(in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the rotation operator is R=1+J_z*dθ, [1+J_z*dθ,d/dt]=0; thus d/dt(J_z)=0, that is angular momentum is conserved