The angular momentum eigenspaces

The eigenfunctions, k defined above are obviously also eigenfunctions of the Dirac matrix (3 (in t ie standard representation), 

Since they are eigenfunctions of and K, they are also eigenfunctions of 

The operator commutes with J, K, and J3, but not with Ho, because it contains the matrix (3 which anticommutes with a. As usual, we denote the eigenvalues of by f(f +1). We have 

Note that the value of i depends on the sign of Kj. For a given Kj, the second basis state belongs to another eigenvalue of L,  

As in nonrelativistic quantum mechanics, the quantum number f has integer values. If f = j — 1/2, the possible values are 0,1,2. in this case we say 

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