Assuming the angular momentum commutation relation, [J_a, J_b] = i hbar epsilon _abc J_c, the eigenvalues of the operator J² are shown to be hbar² j(j+1), where j is an integer or half integer greater than or equal to zero. For a given j, the eigenvalues of J_z are hbar m, where m = -j, -j+1, ... , j-1, j. Thus, there are 2j+1 different J_z eigenvalues for a given J² eigenvalue. The eigenvectors are denoted by |j,m >.