the angular momentum operator L has follow properties:
for a spin 1 operator, we know that it is 3 dimensional. thus:
Thus, we have:
where the size of L is narrowed to l =1 .
with a position bra act from the left, the L becomes:
on the counter part of spin 1 operator, the ket can be replaced by a real vector. and for any vector v, we can expand it by the 3 eigenvectors:
thus, we suitable constant, a unit vector can be written in:
this is how the spherical harmonic enter the vector and then, later in matrix, and become operator.