In three dimensions, this means that we can change our coordinates by rotating about any one of the axes and the equations should not change. Lets try and infinitesimal rotation about the axis. The and coordinates will change.
Now we Taylor expand this equation.
We could have done infinitesimal rotations about the
axes and shown that
all the components of the angular momentum operator commute with the Hamiltonian.
Remember that operators that commute with the Hamiltonian imply physical quantities that are conserved.
Jim Branson 2013-04-22