The quantum orbital angular momentum operators in Cartesian coordinates are derived from the classical angular momentum components

by replacing the position and linear momentum components with their corresponding operators. Since are position components with position operators respectively, and are linear momentum components with linear momentum operators respectively (see eq4), we have

From eq70, we have

Since is defined as the operator for the square of the magnitude of , each of its eigenvalues is the square of the magnitude of the orbital angular momentum of an electron.

###### Question

Can we construct an angular momentum operator using , such that

which is then used to generate eigenvalues?

###### Answer

, a scalar operator, is preferred over , a vector operator, because it easier to manipulate in quantum computations. commutes with its component operators, allowing us to simultaneously determine the eigenvalues of and say, (which is useful, for example in the verification of singlet and triplet eigenstates). It also commutes with the time-independent Hamiltonian , also a scalar operator, implying that we can select a common complete set of eigenfunctions for and . Note that has a form that is consistent with the angular portion of in spherical coordinates (compare eq49 with eq96). An eigenvalue of is the energy associated with the angular motion of an electron, while the square root of an eigenvalue of is the magnitude of the orbital angular momentum of an electron in a particular state.