For a function from a space X to itself, or for an operator that acts on such functions, rotational invariance may also mean that the function or operator commutes with rotations of X. An example is the two-dimensional Laplace operator Δ f = ∂xx f + ∂yy f: if g is the function g(p) = f(r(p)), where r is any rotation, then (Δ g)(p) = (Δ f)(r(p)) -- i.e., rotating a function merely rotates its Laplacian.
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 rotations (in the xy-plane for this example; it may be done likewise for any plane) by an angle dθ the rotation operator is
in other words angular momentum is conserved.