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.

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