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Matrix Representation and Operators

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If the general angular momentum quantum number j is 1 there is a triplet of |j,m_j> states:
|1 ,1>, |1,0> and |1,-1>
In this case a matrix representation for the operators j_x j_y and j_z, can be constructed if we represent the |j,m_j> triplet by three component column vectors as follows
|1,1> =(■(1@0@0)) |1,0> =(■(0@1@0)) |1,1> =(■(0@0@1)) (1)

j_z can then be represented by the matrix

Construct matrix representations for the raising and lowering operators,〖 j〗_+ and j_- acting on the eigenstates |1 ,1>,|1,0> and |1,-1> in the representation given in equation (1)
Use the relationships

j_x=1/2 (j_++j_- ) j_y=1/2i (j_+-j_- )

To construct matrix representations of j_x,j_y and j_z
Show that the matrix representations of j_x,j_y and j_z obey the commutation relation

[j_x,j_y ]=iћj_z

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