Abstract: The angular momentum theory is an important constituent of quantum mechanics. The discrete eigenvalue spectrum of the angular momentum (J², J_z)~ (j(j + 1), m) is determined through the three generators of rotation, as they share the same fundamental commutation relation. Moreover, since under spherical coordinate system J² is identical to the angular part of the Laplacian operator, the angular momentum is thus incorporated into the wave equation, and the wavefunctions for stationary states are believed to be the common eigenfunctions of (H, J², J_z). We notice that unlike wave functions, the state vectors in ket space are expected to carry out all information about the system concerned. When the angular momentum is represented with three sets of independent creation-annihilation operators as it should be, and its action upon the state vectors is scrutinized, it is found that the stationary states are not necessarily the common eigenvectors of (H, J², J_z). By taking the advantage of being separable under both the spherical coordinates and the Cartesian coordinates, the eigenvalue spectrum of (H, J², J_z) for the stationary states of the three dimensional isotropic harmonic oscillator is carefully investigated. For a given stationary state, i. e., with a fixed particle number n which also denotes the energy eigenvalue, while J_z has a discrete eigenvalue, the eigenvalue j (j + 1) can be continuous, thus revealing the true nature of the angular momentum operator J = x × p. For a fixed n , when the m values are sufficiently large, the resulting j (j + 1) is the same as that obtained in terms of eigenfunctions, resulting from the fact that now the stationary states are restricted to a subspace defined by an isoenergetic surface in ket space. Some formulations for angular momentum-related problems should be accordingly readdressed.