SO(3) ROTATIONS

Representation: orthogonal 3X3 matrices that act on 3-d vectors (x,y,z)

Important relationships

_{}

rotation about z

_{}

This shows that any function of φ will be translated to a new function of φ

_{}

Next we work out the commutator
relations and we find

It is easy to see that we get
terms containing second order derivatives and terms with first order
derivatives only. Because of the symmetry of the second order derivatives, all
these terms cancel, and we are left with first order derivatives only. These
terms are due to the fact that the derivative on the left may act on a spatial
coordinate of the second angular momentum factor. We are left with

and this is the z-component of
the angular momentum. Using the fact that the different components are simple
cyclic permutations of eachother we find