Eigenvalue problem for angular momentum in quantum mechanics

In quantum mechanics, an operator acting on vectors ψ in a Hilbert space satisfying the commutation relation J × J = iħJ is generally called angular momentum. The angular momentum and the generators for rotation group share the same fundamental commutation relation, thus the representation |jm> for the generators of SU(2) group, with an eigenvalue spectrum (J², J_z)~(j (j + 1), m), is often taken as the representation for angular momentum. However, such a representation is purely algebraic, while the representation for orbital angular momentum should be formulated on the physical footing since the orbital angular momentum J = r × p is a physical quantity. In the current work, we represent the orbital angular momentum with raising and lowering operators a_k⁺, a_k, k = 1, 2, 3, and tackle the eigenvalue problem for (J², J_z) on the basis of the occupation number notation n₁, n₂, n₃; n_k = 0, 1, 2,… associated with a given spatial reference system. The eigenvalue problem for (J², J_z) can be easily solved by making advantage of the commutativity between J_z and n_b = n₁ + n₂ and that between J² and n_t = n₁ + n₂ + n₃. It is found that the eigenvalues/eigenvectors may reveal a structure regarding to the inherent symmetry of J² or J_z. Our theory can provide a simple and effective methodology for handling angular momentum in quantum mechanics such as in the Landau level problem, spin-orbital coupling, etc., and can be helpful to revealing the inherent structure underlying eigenvalues/eigenstates.