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

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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

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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

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